Math Smart - 7

Some children line up in front of a stall during recess. We cannot see how many
Amazing
children there are altogether because there is a tree in the way. The number of
Mathematician
children in the line is unknown.
In about 825 C.E., the
Persian mathematician and
astronomer Muhammed
ibn Miisa ai-Khwarizmi
developed what is now
known as algebra. He
described his method in a
book which he called Hidab
al-jabr wa'l-muqubala which
means 'Putting together and
balancing up'. The word
algebra comes from the word
"al-jabr" In the title of the
We can say that there are n children in the line.
book.
The letter n is used in place of the unknown number.



















Three more children join the line. There are now n + 3 children in the line.

Think and Share
What if there are two lines of n children?
We say that there are n + n or 2x n children.
How do you write 2
The simplest way to write this is 2n.
groups of n children
and 3 more?
Example 1


A jar contains some marbles, but we don't know how many.
We can use the variable y to represent the number of
marbles in the jar.








By means 3 times of y or
I have three Identical jars of marbles.
3 groups of y.
I have 3xyor3y marbles.






e

I have y marbles and I divide them equally between 4 friends.

Each one gets (y -i- 4) or ^ marbles.

y






Example 2


A banana costs b cents and a watermelon costs w cents. Write down an
expression for the cost of
a) 5 bananas,
b) 3 watermelons,
c) 5 bananas and 3 watermelons
in cents.

b cents w cents
Solution
Cost of:
a) 5 bananas = Sx b = Sb cents
b) 3 watermelons = 3 x w = 3w cents
c) 5 bananas and 3 watermelons = (Sb + 3iv) cents

Example 3




1 don't know. But know
I
Do you know
that his brother, Peter, is
how old 3ohn is?
2 years older than him.



How old is John?
Solution
y + 2 means we add 2 to John's age is unknown. We can represent it using the letter y.
John's age. John is >• years old. So, Peter is (y + 2) years old.

We can also represent Peter's age using the letter z.
z-2 means we subtract 2
from Peter's age. If Peter is - years old, John is {z - 2) years old.



•• Think and Share

The area of a rectangle is the product of its length and breadth. Let
A, .V and _v represent the area of the rectangle, its length and breadth
respectively.
a) Write the formula of the area of the rectangle.
b) If the length of the rectangle is 10 units and the breadth is 6 units,
what is its area?
c) Write the formula of the perimeter of the rectangle.


4aj UNITS introduction to Algebra and Equations

An algebraic expression is made up of constants, variables, operation signs
(+, X, -r), and can include brackets. An algebraic expression has no equal
signs. 3, 7, 4 A term with no
variables is called a
The following are some examples of algebraic expressions.
constant.
4
2n n + iy + y-2x + x,y.z A variable is a changing
3
7
quantity that may have
3
3y -i-2x~w+ 7 y + any one of a range of
possible values. The
letters used are written
term term term in alphabetical order.
3x 0 2y 0 9xy 0 2n, y, vv The parts that are
7
added together are
the terms of the
expression.
coefficient variable coefficient- constant
2, 3, -1 The numerical part
of a term is called its
operation sign
coefficient. It is usually
Example 4 written before the
variable.
Write each of the following as an algebraic expressions.

O Take a number and multiply it by 2. Take another number and multiply it
by 5. Add the two results together. Subtract 10 from the sum.

Solution
Let the two numbers be'a and b respectively. Let us look at each operation.
Think and Share
Take a number and multiply it by 2. 2a
How does it work?
Take another number and multiply it by 5.
Think of a number.
Add these two answers together. 2a + Sb
Add 10.
Subtract 10 from the sum. 2a + Sb - 10 Double the result.
Subtract 6.
Divide by 2.
© Take a number and multiply it by another number. Multiply the result
Subtract the number you
by 13. Add another number to the result. Multiply the result by 3.
started with.
Solution The result will always
be 7.
Let the three numbers be m. n and p respectively. Let us look at each operation.
How does it work? Use
Take a number and multiply it by another number, mxn = mn algebraic expressions to
Multiply the answer by 13. 13 xmn = 13mn explain why.
p
Add another number to the answer. 13mn +
Multiply the answer by 3. 3(13mn+p)








o

Check My
Understanding


O Write an algebraic expression for each question.
a) I have m marbles in a packet. I put another 3 marbles in the packet.
How many marbles are in the packet now?
b) I have m marbles. I lose 7 marbles. How many marbles do I have left?
c) I have 4 packets of marbles. Each packet contains m marbles.
How many marbles do I have altogether?
© A group of 25 school pupils are divided into n groups.
How many pupils are there in each group?

© A pencil costs a cents and a ruler costs b cents.
What is the total cost of
a) 6 pencils,
b) 4 rulers and
c) 6 pencils and 4 rulers in cents?

O Adrian, Cheng and Daylen are playing with some marbles.
Cheng has 4 marbles more than Adrian. Daylen has 5 marbles fewer than Adrian.
If Cheng has x marbles,
a) how many marbles does Adrian have?
b) how many marbles does Daylen have?
O Choose the algebraic expression that matches each part.

a) Take a number and multiply it by 6. Then add another number to that answer.
i) 6x ii) 6x + 6 iii) x + 6 iv) 6% + y v) x + y +

b) Take a number and multiply it by 3. Then subtract another number from that answer.
i) 3^-1 ii) 3p - q iii) 3p - 3 iv) 3p v) p - 3 +
c) Take a number and multiply it by 3. Add another number to that answer, and then
subtract 8.
8
8
i) 3a + 3b — S ii) 3ab - iii) 3a - iv) 3a + 6 - v) a + b -
S
O Decide and assign the unknowns. Then write an algebraic expression for each of the following.
a) Take a number and multiply it by 10. Take another number and multiply it by 6.
Then add the two numbers together.
b) Double the sum of p and 15.
c) Add two numbers together. Multiply it by 15. Then subtract another number
from your answer.
d) Add 1 to half of x.

©* Challenge!
There are x pupils in a class. 22 of them are girls and 15 of them wear spectacles. Write down an
algebraic expression for each situation below.
a) The number of boys in the class.
b) The number of pupils not wearing spectacles in class.




Introduction to Algebra and Equations

3.1.2 Simplifying algebraic expressions



In algebra, we can simplify long algebraic expressions, just like in fractions, by
grouping the like terms. What are like terms? ^ RECALL

Like terms In Stage 3, we learnt to
identify like fractions to
Like terms are terms that have the same variable(s) and each variable must
compare them.
have the same power/index.
and ^ are like
When two terms are not like terms, they are known as unlike terms.
fractions.
Sx + 5y + 7a: = 3x + 7x + 5y
Fill in the blank.
= lOx + 5y
Like terms Unlike terms
2x + 7x 2x + 7b
6m — 15m 6m — 15m^
14xy — 12xy 14xy - 12y
25a^ + 12a^ 25a^ + 12a^
4y + 7y 4y + 7
Think and Share


Example 5 Are 14xy and 12yx like
terms? Why?
Simplify the expressions.
x
a) 4a + 7a b) 9x - c) 3t — 25
d) 8a + 7b + 2a - 5b e) 3x + 4 - 2x -
f) 8a + 9b + 10c - 5 + 8b - 2a - 6c

Solution
a) 4a + 7a = 11a

b) 9x - X = 9x - Ix
= 8x X is the same as Ix

c) 3t-2s The terms are unlike terms so we
cannot simplify this expression
d) 8a + 7b + 2a - 5b = 8a + 7b + 2a — 5b Find the like terms Note that the sign in
front of each term stays
= 8a + 2a + 7b - 5b Group the like terms
with the term.
= 10a + 2b
t
= 2(5a + b) Simplify











o

e) 3x + 4 - 2x - 8 = 3a: + 4 - 2x - 8 Find the like terms
= 3x - 2a: + 4 - 8 Group the like terms
= 1a:-4
= X - 4 x is the same as 1a:

f) 8a + 96 + 10c - 5 + 8b - 2a - 6c

= Ba - 2a + 96 + 86 + 10c - 6c - 5 Group the like terms
= 6a + 176 + 4c - 5 Simplify


Example 6


Complete the algebraic pattern.






5x 4x 2x

Solution
To find the expression
in each block you add Sx + 4x = 9x 9x 6x 4a: + 2a: = 6x
the expressions in the
5x 4x 2x
two blocks below it.
15x 9x + 6x= 15x
9x 6x
5x 4x 2x

Check My
Understanding


O state true or false for each statement. If the answer is false, give the correct answer.
a) 2a + 5a = 6a b) 7x + 5x = 12x
c) 2d+ 5d+ 8d= 15d d) 14t + 7t-5=16c
e) 2a + 36 = 5a6 f) 96 + 6 + 46 -2= 136 +4
O Simplify these expressions where possible,

a) 5a + 4a b) 8d -3d
8
c) 2a + 36 d) 6a + 2a +
e) 9ki ~ 3ld f) 8t + 7t + 85 + 25
g) 12^ + 4g + 6k - 2k h) 13a + 2 + 2a-l
i) 13a6 — 3a + Sab j) 7xy + 2x + 3y - 2xy + y - 3y +

k) 3a6c + 66c + 7b — 46c + 9-26 I) 4y + 16 + 5y + 2xyz - 9 — xyz
m) 16mn + 30 + 15m - 12mn - 15 - 3m + 2n — 3mn
n) 5x + 2Sxy + 6 + JSxyz — 2x + 14 - 3x + lOxy



52^ UNIT 3 I Introduction to Algebra and Equations

O Write an expression for the perimeter for each shape.
Express your answers In simplest form.

a) b)










O Complete the algebraic patterns.

a) b)
12a
Sa 11 j

2a Sy 4y
c) d)
15a: + 25
5x+2
b
1
2a + 5h 3a+4h 5a+ 3.V +

O The diagram shows some expressions that are equal to 6x + 5y.

Find six more expressions that are equal to 6x + 5y.


10.V + 5v-4x



6x + Sy



6a- + 13v-8v



O Complete the algebraic pattern.


Ma + 6b+ 3
7
8a + b +
Sa+h-2
3a + 2b-S

<p Expanding algebraic expressions

Q Write each The distributive law states that to expand an algebraic expression means to
multiplication multiply each term in the bracket by the term outside the bracket.
sentence as repeated
5(Ar + 3) means '5 groups of (x + 3)'.
addition.
(a) 2 X 2 = So, 5 (a: + 3) = (x + 3) + (x + 3) + (x + 3) + (X + 3) + (a: + 3)
= 5x + 15.
(b) 6 X 3 =

(c)4x(3 + l) = 3 (x + 4) = (3 X x) + (3 X 4)
= 3x + 12

When you expand an 7 (2a - 3) = (7 X 2a) -(7x3)
expression, you multiply = 14a - 21
each term in the brackets
This is the distributive law where 3 groups of x + 4 is the same as 3 groups of x
by the terms outside the
and 3 groups of 4.
brackets.
The rules by which operations are performed when an algebraic expression
involves brackets are the same as in arithmetic.
• Simplify the expression within the brackets first.
• Use the distributive law when an algebraic expression within a pair of brackets
is multiplied by a term.

Example 7


Expand the expressions.

a) 3(a + 5) b) 2(5x-2y)
c) -2(y-5) d) 5(x + 7) + 6(3x - 5)

Solution

a) 3(a + 5) = (3 X a) + (3 X 5) Multiply each term in the bracket by 3
= 3a + 15
When the bracket is b) 2(5x - 2y) = (2 x 5x) - (2 x 2y) Multiply each term in the bracket by 2
preceded by a + sign, = lOx - 4y
the signs within the
c) -2(y-5)=(-2xy)-(-2x5) Note that-(-10) is 10
bracket do not change.
= -2y + ia
When the bracket is
d) 5(x + 7) + 6(3x - 5)
preceded by a - sign,
= (5 X x) + (5 X 7) + (6 X 3x) -(6x5) Expand each set of brackets
every sign within the
bracket changes: = 5x + 35 + 18x — 30
= 5x + 18x + 35 - 30 Group the like terms
+ to -
5
= 23x +
-to +






UNITS Introduction to Algebra and Equations

Check My
Understanding


O Choose the correct answer.
a) 2(a + 5)
i) 2a + 10 ii) 2a + iii) 7a iv) 12a
5
b) 7(6 - 4)
i) 76-4 ii) 7p iil) 36 iv) 76 - 28
0 12(x + 5)
i) 12a: ii) 17a: iii) 12x + 60 iv) 12x +
5
d) 5(2/c-l)
i) 9k ii) 10k - iii) lOfe iv) lOfc
5
e Expand the expressions.
a) 2(x + 3) b) 3(6 + 5) c) -6(2+y) d) 3(a:-2)
e) -9(2-c) f) 5(12-/c) g) -Qixy + 1) h) -3(4c - 2)

O Expand the expressions,
a) 3(2a + 36 + 4) b) 4(3a: + 2y + 1) c) -2(4m + 5n-3)
d) 5(6x-4y-7) e) 7(5a - 36 + 5c) f) -6(7c + 2d - 4/)
O Expand and simplify the expressions.
a) 8(77i + 2) + 5(m + l) b) 4(5 + 36) + 6(2 + 66) c) 6(2 - 3y) + 5(6y - 3)

d) 4(7-3x)+5(2;c-3) e) 6 + 5(2x-3) + 12 f) 2a + 3(5a + 6) - 12 + 4a



Investigate!


Instructions
O Start with a number.
@ Add 5.

O Multiply 3.
O Subtract 15.
O Divide by 3.
O Write down the answer.

If you start with the number 10, you get: 10 •15 45 30 •10
a) Using the instructions above, find the answers for the following.









b) What do you notice about your answers in a)?
c) Write down the algebraic expression that you think represents this
set of instructions.

CHAPTER 3.2

In this chapter
eriving and Using

Pupils should be able to:

• substitute positive
integers into simple Formulae
linear expressions/
formulae
• derive and use simple
3.2.1 Substitution
formulae

Substituting into an algebraic expression means replacing the variables with the
given numbers to evaluate the expressions.


Example 1


\f X = 2, y = 3 and z = 5, find the values of the expressions.

a) 3x + b) 5 + 3y c) xz — yz
z
Solution
z
a) 3x + b) 5 + 3y xz — yz
= (3x2) +5 = 5 + (3 X 3) = (2x5) -(3x5)
5
= 6 + = 5 + = 10-15
9
= 11 = 14 = -5
Check My
Understanding


O x = l and y = 5, find the value of

a) ;*: + 5 b) x — y c) 3x d) 2x + 3y
e) Ay-2x f) x(3 + y) h) Ji -y
g) £
7
If
@ a = 2 and b = S, find the value of each expression.
a) a + b b) 3b c) a — b d) b — 3a
e) ^ f) 1 - f q) 3fl + 2/7 h) ^ - 3fo
b 2 ^ ^
@ p = 2, q = 4 and r = 1, find the value of each expression.
\f
a) p + q — r b) 3p + 2q + 5r c) 2p + q - 4r d) 2(p + q-5)
e) p(r + q) i) H g) Q - h) M
2 ^ 4 ^ 3
O When a = 3, b = 4 and c = 5, what is the value of each expression?

a) 2(abc) b) 0 c{ab) d) a^b
2


Introduction to Algebra and Equations

Answer the following.
a) Find the odd one out when m = 7.
i) m + 5 ii) 21-m iii) 2m-2
b) Find the value of a that makes all of these expressions equal,
i) 2a + 1 ii) 10-a iii) 3a-2
c) Find the values of x and y that makes all of these expressions equal,
i) a: + 2_v + 16 ii) 2x + 2_v + 15 iii) x + 3_y + 11
o y = 4 z = 3
Write two expressions in y and s that have the value of
a) 16, b) 20, c) 19.
Is each pair of expressions equal? Use substitution to explain your
answer.
a) 3x and x + 3 b) x^and 2x c) 4x^and {4x)^



3.2.2 Working with formulae



We construct a formula to find the solution with a range of values to any word
problem. Choose the first letter of each quantity to represent the quantity before
expressing the rule in algebraic terms.

Example 2


O Adela is 8 years older than Olivia.
a) Write down an expression connecting Adela and Olivia's ages in words.
b) If Adele is A years old and Olivia is 0 years old, write a formula
connecting A and 0.
@ A book costs $12 and a magazine costs $8.
a) Work out the total cost of 5 books and 2 magazines.
b) Work out the total cost T in dollars of b books and m magazines.
O The formula for converting /itres to millilitres is to multiply the number of
litres by 1000.
a) Use the letters in red to write the formula using variables.
b) Use the formula to calculate the number of millilitres in 1.2 litres.

Solution
O a) Adela's age = Olivia's age + 8 years
8
b) A = 0 +
© a) Total cost = 5x$12 + 2x$8
= $60+ $16
= $76

b) T = bx 12+ mxB
T = 12b + 8m
O a) Number of millilitres = 1000 x number of litres
m = 1000 X /
b) m = 1000x 1.2
= 1200 millilitres
e

Check My
Understanding



O a) How many hours are there in 3 days?
b) How many hours are there in 7 days?
c) Write down a formula for the number of hours In a days.
O A small bus can carry 13 passengers. A large bus can carry 60 passengers.
a) How many passengers can be carried by 1 small bus and 1 large bus?
b) How many passengers can be carried by 5 small buses and 3 large buses?
c) Write down a formula for the total number of passengers, P, which can be carried
by .V small buses and y large buses.
0 The formula for changing minutes, m, into hours, h, is h =
Use the formula to find h when
a) m = 120 minutes b) m = 300 minutes c) m = 45 minutes
O The number of hours in a given number of days can be found by multiplying the
number of c/ays by 24.
a) Use the green letters to write the formula using variables.
b) Use the formula to calculate the number of hours in 14 days.
0 The number of classrooms needed in a school can be estimated by dividing the
number of pupils in the school by the average number of pupils in the class.
a) Use the green letters to write the formula using variables.
b) Use the formula to find the number of classrooms needed in a school of
300 pupils that has an average of 30 pupils in a class.
O The perimeter of a rectangle is found by adding its /ength to its width and doubling
the answer.
a) Use the green letters to write the formula using variables.
b) Use the formula to find the perimeter of a rectangle that is 7 cm long and
2 cm wide.
0 The average speed of a runner can be found by dividing the distance covered by the
time taken.
a) Use the green letters to write the formula using variables.
b) Use the formula to determine the average speed of a runner who covers
80 metres in 10 seconds.
If
0 you count the number of seconds between a flash of lightning and the noise of
the thunder and multiply this by 3, you get the distance in /cilometres to the centre
of the storm.
a) Use the green letters to write the formula using variables.
b) Use the formula to work out how many kilometres away the centre of the storm
is if you count 15 seconds between the flash of lightning and the noise of the
thunder.







I UNIT 3 I Introduction to Algebra and Equations

O P = 3;c-1,Q = 4-^andR = 5j-8
Use the values of P, Q and R to expand and simplify the following expressions,

a) 3P + 2Q b) 4P + 3R c) 2P + 3Q + 4R d) 5P + 4Q + 2R-6
Challenge! 2;c-1
a) Write an algebraic expression for the perimeter of the
quadrilateral in simplest form. y
b) Find the perimeter when = 3 cm and y = 4 cm.
c) When = 1 m and v = 0.5 m, which of the four sides Is the longest?
2x + 3




Investigate!
































O Leia wants to hire a bicycle. The table shows the cost, C, in dollars, of
hiring a bicycle for.v hours from Company M and Company N.

Company CostC($) .
M C = Zv +
5
N C = 3a- +
1

a) Find the cost of hiring a bicycle for 1 hour from each company.
Which company is more expensive?
b) Find the cost of hiring a bicycle for 6 hours.
Which company is more expensive?
c) For which value of .v will the cost of hiring a bicycle from the two
companies be the same?
d) Leia wants to hire a bicycle for 8 hours. Which company should she
choose? Give a reason for your answer.




©

CHAPTER 3.3


In this chapter
Functions a Mapping
Pupils should be able to:
• represent simple
functions using words,
symbols and mappings 3.3.1 Functions
• use letters to represent
unknown numbers or A functional relationship exists between paying for petrol and the litre. The
variables more litres of petrol we buy, the more we have to pay. We can find the value
of one variable {such as the petrol price) when we know the value of the other
variable (the number of litres needed).
The relationship between two variables in which each value of the first variable
corresponds to exactly one value of the other variable is called a function. It is
like a function machine that performs one or more mathematical operations on
a number entered to produce an answer.


Check My Input
Understanding This function machine
adds 2 to any number (1) to
give the number (3) that comes
Write down the function
out of the machine.
in the function machine.
Output


The numbers that are put into the function machine are called the input
numbers. The numbers that come out of the function machine are called the
output numbers.
Some function machines can perform more than one operation.

This function machine takes
in a number (5) and multiplies input
Output
by 2 to get 10, and then
adds 3 to get 13.

So, the output is 13 when the
input is 5.
Output


Input Output
We can draw a map
to show the function 4
'multiply the input 7
number by 2 and
8
then add 3'.
10





UNITS Introduction to Algebra and Equations

.3.2 Mapping diagrams



A mapping diagram shows a function. Input Output
4 \
The following mapping diagram tells us that U) >. 11
4 maps onto 11, 7 — 17
7 maps onto 17,
8 >■ 19
8 maps onto 19,
10 23
y
J
10 maps onto 23 in this function. V ^
The function that maps the input values onto those X -^2x + 3
output values is written as a: ^ Ix + 3.
We say that: 'x is mapped onto Ix + 3'.

Example 1


Find the output number of this function machine.


6 x3 +
5

Solution
We can use the function machine to do the following calculations.
6 X 3 = 18
18 + 5 = 23
So, the output number is 23.

Find the input number of this function machine.


+ 2 -3 6



Solution
To find the input number, we write the inverse operation below the
boxes.
The inverse of 'dividing by 2' is 'multiplying by 2' and the inverse of
'subtracting 3' is 'adding 3'.





>-<±i



18 2 -3 0
xT)<—(TI




So, the input number is 18.




o

Check My
Understanding

O Foi" each function machine, calculate the output.

a) , , b)
12 ^4 25^-6 —>►? 82

d) e) f)
x4 + 7 -? 15 -^3 -6


o For each function machine, calculate the input,

a) b) C)
X 10 -5 -65 ?- ^3 + 8 38 x4 + 3


O Calculate the missing number in these function machines

C)
5^ X + 7 -^ 27 32^ : 4 + 58 12^ -2 ^ + ^


O Por each function machine, calculate the missing values^
a) 5 f



24
-2 x3

36





b)


+ 9 -^3



1


O Write down the function in each function machine,
a) b)
5 , -^30 -2



-^.36


42 ■^-2




6^1 UNITS I Introduction to Algebra and Equations

O Complete the mapping diagrams,
1
a) A" —7a - b) A—>3(a + 1) c) .j: A +
3
Input Output Input Output Input Output
N
f \ f' N, ^
53 5 3

9 27 6
81 12 13
15 78 12
L J k J k J V J k y



Investigate!



Cheng, Mila and Reza are asked to write down the function of this
function machine.



a) 5


13 21



19 36

Cheng says, "I think the function is 'multiply by 2 and then subtract 2"'.

Mila says, "I think the function is 'multiply by 2 and then subtract 5"'.
Reza says, "I think the function is 'subtract 3 and then multiply by 2"'.
Who is correct?
O Find the function for each of these mappings.
Write each answer in the form 'a —>■

a) b)
Input Output Input Output
\
^
r N
20 15 6
21 16 5 10

22 17 4 14
23 18 5 18
J
J
k ) L V k
J





o

CHAPTER 3.4


In this chapter onstructirf^ and

Pupils should be able to:
• construct simple linear
Solving Equations
equations with integer
coefficients
• solve linear equations
where unknowns are on
We form equations and solve them in many situations. We can use equations
one side only
to calculate a phone bill, plan a fundraising event at school or to work out the
price at which I should sell my products to earn a profit if I open a shop.

When we give a value to an algebraic expression, we get an equation. Let us
learn how to formulate and solve equations.



3.4.1 Solving equations using trial


and error


We can use trial and error to determine the value of x that satisfies the
equation.


Example 1



Amazing O x + S = 7
Mathematician Solution
Substitute values for x into the equation such that the value on the LHS is
I The equal sign (=) was
equal to the value on the RHS.
invented nearly 500
years ago by the Welsh If X = 1, LHS = 1+5 = 6. LHS RHS, so x = 1 is not a solution of the
equation
mathematician Robert
Recorde. If X = 2, LHS = 2 + 5 = 7. LHS = RHS, so x = 2 is the solution of the
He invented it because he equation.
was tired of always having If X = 3, LHS = 3 + 5 = 8. LHS 9^ RHS, so x = 3 is not a solution of the
to write "is equal to" in
equation
his equations.
So, X = 2 satisfies the equation x + 5 = 7.
He chose the two lines
because "no two things
X * + 5 Is it equal to 7?
can be more equal".
1 1 +5 = 6 X
2 2 + 5 = 7 3
3 3 + 5 = 8 X








UNITS I Introduction to Algebra and Equations

Check My

Understanding


O For each equation, does the solution given in the bracket make the equation true?
Answer Yes or No.
a) y - 3 = 4 (y = 8) b) 6x = 18 {x = 3) c) 1=8 (/"^IS)

-4
f)
X
d) 3a: + 2 = 8 (a: = 2) e) j-2 = 5 (a: = 9) |= 1 (d = 25)
O Choose the correct solution to each equation,
a) 3x - 36
i) a: = 6 ii) a: = 12 iii) a: = 33 iv) x = 11 v) x = 13
4
b) f =
8
i) a: = 3 ii) X = 6 iii) X = 1 Iv) X = 0 v) X =
7
c) 6x - 5 =
3
1
S
1) X = ii) X = iii) X = 2 iv) X = 6 v) x =
d) f + 6 = 8
i) X = 5 11) X = 1 iii) X = 9 iv) x = 6 v) x = 12

O Challenge! Find the value of x for each equation.
9
2x+ 1 b) ^ + 6 = c) iM.3
1
a) = 3 d) -7 =
J


Solving simple equations



Think of the balance as an equation. It represents the equation x + 2 = 5. To find
the value of x, you need to find a way to leave x on its own.



. 6)nn.






If we take away 2 from both sides of the balance, we are left with x = 3.
We need to keep the scale
x + 2 = 5 balanced when adding,
x + 2 — 2 = 5 — 2 Subtract 2 from both sides of the equation. subtracting, multiplying
or dividing terms in the
X = 3
equation.
n □□□ -

So, the solution of the equation \sx = 2.
To solve equations,
In other words, .v = 3 satisfies the equation a + 2 = 5.
you use the inverse
operation. We can also use number pairs to help us find
-2 is the inverse of +2 the solution. So, a = 3

We can use substitution to check if the value of the variable is correct.

Example 2

Solve the following equations for x. Then check your answers.

O a + 8 = 25 Solution

A + 8 = 25
x + 8- 8 = 25-8 Subtract 8 from both sides of the
/ LHS = 17 + 8 = 25
So, a: = 17. equation (- 8 is the inverse of + 8)


@ a-7 = Solution
9
a-7 =
9
a-7 + 7 = 9 + 7 Add 7 to both sides of the
So, a = 16. equation (+ 7 is the inverse of - 7) / LHS = 16-7 = 9

© 3a: = 24 Solution

3x = 24
Divide each side of the equation
3 3 by 3 (the inverse of multiply by 3
So, X = 8. is divide by 3) / LHS = 3 X 8 = 24

© ^=7 Solution
.Y
5 =7
5 xi = 5 X7 Multiply each side of the equation
/ lHS= 35 = 7
So, X = 35. by 5 (the inverse of divide by 5 is ^ 5
multiply by 5)

Check My
Understanding


O Solve the equations by finding the values of the unknowns.
a) X + 9 = 52 b) a + 7 = 16 c) y-8 = 2
d) z + 10 = 25 e) 15 = q - 11 f) 40 = b + 25
O Solve the equations.
a) 5x = 30 b) f =12 c) 6y= 18
d) y = 12 e) 48 = 12z f) 20 = .'
-i\
Find the value of each unknown,
a) m + 15 = 30 b) 12 = a-5 C) d) 64 = 8m
* =2
e) ^=7 f) g) 12q = 108 h) 75 = c + 25



UNITS Introduction to Algebra and Equations

For Questions 4 to 1, write down an equation. Then solve the equation.
O Mary thinks of a number and adds 5. The answer is 14.
What is the number?
O John thinks of a number and multiplies it by 6. The answer is 42.

What is the number?
O Amina thinks of a number and subtracts 14. The answer is 18.
What is the number?
O Steven thinks of a number and divides by 8. The answer is 12.

What is the number?



3.4.3 Solving equations with two

operations



In this section, you will learn to solve equations that involve two operations.

Example 3


Solve each of the following equations. Then check your answers.

O 2;c + 5 = n Solution

2x + 5 = 11
2x+ 5- 5 = 11-5 Subtract 5 from both sides of the equation / LHS = (2 X 3) +
5
2:t = 6
= 6 + 5
h =A = 11
2 2
x = 3
O £^ = Solution
3
3
=
^
4
4x(lzl)=4x3 Multiply both sides of the equation by 4
4
17-5
x-5 = ^2 / LHS =
;c-5 + 5 = 12 + 5 Add 5 to both sides of the equation 12
= 4
x = 17
= 3
O 17-5a: = 7 Solution
17-5x = 7
17 - 5;*: + 5x = 7 + 5a: Make x positive by adding 5a: to
^ both sides of the equation
17 = 7+ 5a:
17-7 = 7 +5a:- 7 Subtract 7 from both sides of the equation
10 = 5x
/ LHS = 17-{5 X 2)
Y = ^ Divide both sides of the equation by 5
= 17-10
x = 2 = 7


o

O 5x + 1 = 7 Solution
5x + 1 = 7
5x + 1 -1 = 7 - Subtract 1 from both sides of the equation
1
5x = 6
Divide both sides of the equation by 5
/lhS = 5 X I +1
= 6 +
1
= 7


Investigate!


PAIR WORK
The mind reader game
The result obtained at the end of this game will tell you their age and
the month in which they were born.
Instructions
O Start with the number of the month In which you were born. (January is
1, February is 2, March is 3, etc).
@ Multiply this number by 4.
@ Add 10.

O Multiply by 25.
@ Add your age.
O Subtract 365 (the number of days in a year).
O Add 115. What is your final answer?

You should find that the last two digits of your answer are your age, and the
other digits represent your birth month.
Now try this on a friend to check if it works. Discuss the trick with them.
Questions
O Can you explain why it works using algebraic equation?
O Show your working clearly.


Check My

Understanding

O Solve these equations for x. Show your working clearly. Then check your answers by
substituting your answer into the equation to see if it is correct.
a) 2a: + 8 = 20 b) 5a: + 12 = 27 c) 4x - 3 = 9
d) 6x - 5 = 19 e) 0 = 4x - 15 f) 23 - 5x = 13
g) 100-7x = 44 h) f + 3 = 9 i) 7 - 12 = 18

A--3 6 +
A*
j) 9 + 2 = 10 k) = 15 I) 1 =



68 ( UNIT 3 I Introduction to Algebra and Equations

o Solve for x. Then check your answers by substituting your answers into the equation to see
if it is correct.
a) X + 3 + 4x = 33 b) 3x + 4 + 2x + 1 = 45
c) X - 3 - 7x + 5 = 10 d) 20 = 2x - 4 +
X
Write down an equation to represent each problem. Then solve each equation.
a) I think of a number and multiply it by 3. 1 then add 5.
The answer is 17. What is the number?
b) I think of a number and multiply it by 12. 1 then subtract 7.
The answer is 77. What is the number?
c) I think of a number and subtract 6 from it. The answer is 15.
What is the number?
d) I think of a number. The product of this number and 15 is 30. What is the number?
e) Paul has some sweets. Jane has 5 sweets. Together they have 19 sweets. How many
sweets does Paul have?
f) There are 28 pupils In a class, 15 of them are girls. How many boys are there in the class?
g) The sum of two numbers is 30. One number is 14. What is the other number?

o Solve the equations.
8
a) 14 = 2(x-8) b) 3(x + 1) =
c) 0 = 4(;3x-2) d) 5 + 3(x-2) = 23
e) 67 = 3 (4x - 3) - 8x f) 3(x + 4) + 2(x - 5) = 32

Write down equations and solve them to find the unknown quantity in each problem.
a) The length of wire fencing needed to enclose a rectangular
vegetable garden is 13 m. The length of the garden is 4 m.
Remember that
Find the value of w, the width of the garden.
the perimeter of
b) The perimeter of a square floor tile is 84 cm. The length of the floor
a shape is the sum
tile is X cm. Write an equation for the perimeter of the tile and
of all its sides.
use the equation to find the length of the floor tile.
c) The perimeter of a rectangle is 30 cm. Its length is twice its breadth.
If the breadth is x cm, find the value of x.
d) Hannah and her 4 friends share a bag of 17 sweets. There are 2 sweets left over.
If each person receives x sweets, write an equation for the total number of sweets in the
bag. Use the equation to work out how many sweets each person receives.
k.



















e

Iievision



O Work out the values of the following Q Write an equation for each problem and
expressions using: a = 2 and b = A. then solve for .v.
a) aft b) a) Amanda thinks of a number. She adds
ft 2 to the number and then divides it
c) d)
a by 4. The answer is 3. What is her
e) 3(a + ft) f) ab — Sb number?

Q Simplify the following expressions where b) Farhana thinks of a number, multiplies it
possible. by 3 and then adds 7. The answer is 25.
What is the number?
a) 9c - 4c
b) Sab + ISab 3 Write down an equation to represent the
c) 3m + 14p + 6m - 8p following problem. Explain why the answer
to the following is always 4.
d) 9xy + 6 + 4x
7
e) 12cd + 8 - llcd - Operations Algebraic Expressions
f) 2xy + Sx + 6yx - 4:c + 9 + 4y + Bz — 6 + Add 7 to the number.
4xyz - 4y + xyz Double your answer.
Q Simplify.
Subtract 6.
b
a) 4 X ax b b) 3a + a +
Divide by 2.
c) 8xb-bx8 d) S{b + 3)
Subtract the number
e) 3(x - 4) + 2x f) 6a + 2a(3 + 4ft) you started with.
O Write an algebraic expression for each of @ Solve each equation and write your answer in
the following.
the puzzle.
a) The number of days in x weeks.
b) The cost of apples if one apple Across
costs 75 c.
1. f=^
c) The number of cents in y dollars.
2. 6a = 12
d) The selling price of a shirt if the cost
CO 3. a-10 = 2
price is $a* and the discount is $y.
II 5. I +9 = 16
e) The sum of twice a number and 3 times
7. a-2 = 14
of another number.
8.
3 Solve for X.
9. a-7 = 16
a) 2x = 18 b) X + 7 = 10
11. 4a + 2 = 50
c) d) x-4 = 11 12. The value of 5a + 10 when a = 9.

13. The value of 2a + 2 when a = 12.
e) 4x + 7 = 19 f) 5x-2 = 20
X X - 8 o Down
g) 3 +5 = 10 h)
6
1. a-8 = 7
i) 2 + 3 (X - 4) = 14
2. a+ 4 = 28
j) 2(5x - 3) - 6x = 22
4. ^ =10





UNIT 3 I Introduction to Algebra and Equations

6. f=5
7. 3a - 11 = 28

8. 2-7 = 2
9. 3a + 5 = 80
10. ^+5 = 16 10

11. 5a - 20 = 60
11


13
n







Mathematics Connect



Linear algebra is the branch of mathematics that • If you want to find out what a particular gene
deals with linear equations like the ones we have actually does, you have to see how it Influences
dealt with in this unit. all the chemical processes in our body. The way
these processes work, and how they influence
Economics
each other, can be expressed by large systems
Economists use linear equations to plot the supply of linear equations.
and demand of a particular product in order to
• A biologist uses linear equations to get an idea
make decisions about prices and distribution of
of how a population of animals might change
goods. They can also use the results to predict the
overtime.
economy or the future profits of a company.
• An engineer will use linear equations to work
Computers out the exact proportions of a building like a
Computer graphics programmers used linear bridge or a high-rise building, and how much
algebra in the creation of Google. and what kind of materials to use.
Linear algebra is widely used in coding theory. It As you can see, equations are a fact of life for many
is used to encode data in such a way that if the people and to be able to work with them, you need to
encoded data is tampered with a little bit, the
start with the simplest ones — the linear equations.
original data can be recovered.
Most processes involved in computer graphics
such as rotation, scaling and perspective are
implemented using linear algebra.

Other areas where linear equations are used
Linear equations are the simplest equations there
are. Equations are used in many other areas to
model the world around us.
• All of quantum computing is literally just linear
algebra, as is general quantum mechanics.

Help
Sheet




Algebra uses letters and symbols to represent words and numbers.
Variables represent an unknown value or a range of values.
A constant is a term with fixed value and is usually written as a numeral.
A coefficient is the number in front of a variable (ie. 5y, 22abc, etc).
Expressions represent unknown values in algebraic form. They can be simplified but the value
of the variable cannot be found.
Equations contain an equal sign to show that the LHS expression is equal to the RHS expression.
The values of the variables can be found by using the inverse operations. The values can be
substituted back into the equations to check if they are correct.




Working with algrebraic terms
• Variables are written in alphabetical order (ie. 25mnp, 1 ^xyz, etc).
• Coefficients are written in front of the variables (ie. 7xy, - ^OcJef, etc).
• When the coefficient of the variable b is 1, we write it as b, not ^b.




Working with an equation step-by-step



Remove the brackets Group like terms Isolate the variable

3{x + \) = 2x + 2A-X 3x + 3=(2x}+ 24 Q) 3x + 3 = X + 24
- X -
X
3x + 3 = 2x +24-X 3x + 3 = X + 24
2x + 3 = 24


f


Remove the constant Remove the coefficient Check your solution
2x ^ 21 21
2x + 3 = 24 3(^ + 1) = 2(^) + 24- 2
2 2
-3 -3
x = ^ or 10.5 3(£i+ |) = « +
2x = 21 2 2' 2
2
^2^" 2 2


2 2









UNITS Introduction to Algebra and Equations

Decimals





Ancient civilisations had their own numeral systems. The ancient Egyptians used the
decimal system (base-10), the Babylonians used a base-60 system. Some civilisations used
a base-20 system. We use a base-10 system called the decimal system or the Hindu-Arabic
System. The decimal system uses 10 numerals-0, 1, 2, 3, 4, 5, 6,1, 8 and 9. All numbers in
the decimal system are made using a combination of these 10 numerals. It is thought that
the decimal system may have been based on counting with our 10 fingers.

The ancient Egyptian number system was also a base-10 system. The table below shows
the symbols used in the ancient Egyptian number system and the symbols used in the
decimal system.





I n e. \ %





stroke heelbone coiled rope lotus flower pointed finger tadpole scribe
1 10 100 1000 10 000 100 000 1 000 000



You will learn about:
In this
Understanding decimals
unit
Ordering decimals
Rounding decimals

Adding and subtracting decimals
Multiplying and dividing decimals
by 10,100 and 1000
Multiplying decimals

Dividing decimals

Estimating and approximating



Try and Apply!


I Use the ancient Egyptian
-j
; number system symbols to
;^| represent the number 123.

CHAPTER 4.1

In this chapter
Understanding
Pupils should be able to:
• interpret decimal

notation and place value
Decimals


^ RECALL We see and use decimals in our everyday lives. Your weight on a digital scale
may show 32.53 kg, and a receipt may show $204.20. Do you know how much
O In the decimal 5.63, money you need to hand over based on the receipt?
what do the digits 5,
Decimals can show values between two consecutive whole numbers. It is
6, and 3 stand for?
another way to represent quantities that are parts of a whole so as to improve
State their values.
the accuracy of the quantity.
Q How do you write
A decimal point separates the whole number from the fraction part of the
36 hundredths
number.
in fraction and
decimals?
' Think and Share
Q Write 345 cents in
dollars.
In many sports, the measurement of time needs to be as accurate as
possible. For this reason, sports timing is usually recorded in hundredths of
a second.
In 2009, Usain Bolt broke the men's world record for the 100 metre sprint.
His time was 9.58 seconds.
If an athlete ran a faster time, even if it was only one hundredth (0.01) of
a second faster, they would become the new record holder.

How many decimal places are used to measure Usain's 100 metre sprint?






Decimal notation

Spotlight A decimal Is a fraction whose denominator is a power of 10.


0.4 (4 tenths) =
Numbers within the
decimal system may
• be whole numbers 0.09 (9 hundredths) =
100 decimal point
(e.g. 327)
A decimal point is used to separate the whole number
• contain parts of a
from the fraction part of the number. 327! 251
whole (e.g. 327.25)







UNIT 4 Decimals

Place value


The place value of a digit depends on the position of the digit in a number.
Let's look at the place-value table.










1 2 9 4 • 7 6 5




The value of The value of The value The value The value of The value of The value of
the digit 1 is the digit 2 is of the digit of the the digit 7 is the digit 6 is the digit 5 is
1000. 200. 9 is 90. digit 4 is 4. 0.7. 0.06. 0.005.
1 thousand 2 hundreds 9 tens 4 units 7 tenths 6 hundredths 5 thousandths
1000 200 90 4 J_ 6 5
10 100 1000


The expanded form of the decimal 1294.765 is

1 X 1000 + 2X 100 + 9X 10 + 4X1 + 7X:jL + 6X:^+5X:|^

+
= 1000 + 200 + 90 + 4 + i
= 1000 + 200 + 90 + 4 + 0.7 + 0.06 + 0.005




Check My
Understanding


state whether the following statements are true or false. If it is false, write the correct
statement. In the decimal 489.26,
a) 489 is the whole number.
b) The decimal point separates the whole number of the decimal number from the
fractional part of the decimal number.
c) There are 2 hundredths.
d) There are 6 tenths.
e) The place value of the digit 4 is hundredths.
What is the place value of the underlined digit in each of the following numbers?
a) 39.4 b) 39.04 c) 169.08 d) 1473.98
e) 319.473 f) 105.24 g) 286.449
What is the value of the underlined digit in each of the following numbers?
a) 39.4 b) 39.04 c) 169.08 d) 1473.98
e) 319.473 f) 105.24 g) 286.449





e

CHAPTER 4.2


In this chapter
Operations on
Pupils should be able to:
• add and subtract
integers and decimals, Decimals
including numbers with
different numbers of
decimal places
The One World Trade Center in New York is 546 m tall. How much taller is it
than the Eiffel Tower in Paris? To answer this question you add or subtract
whole numbers. Sometimes, you need to add or subtract decimals. For
example, what is the height difference between Burj Khalifa in Dubai, and
the Lotte World Tower in Seoul?
In this chapter you learn how to add and subtract decimals.
Burj Khalifa
829.8 m



Lotte World Tower
554.5 m
One World Trade
Center. 546 m
Empire State
Building|443 m
Eiffel Tower Burj Al Arab
324 m 321 m
The Shard
310m
Taj Mahal
73 m





O Spotlight



Take a tour around the
world and look at some
of the tallest skyscrapers
that are still under
construction.

SiaiH











UNIT 4 I Decimals

4.21 Adding and subtracting decimals recall



When you add and subtract decimals, make sure that the decimal points are lined
O Find the sum.
up. You can use a place-value table to help you align the digits in the correct
1.08
place-value columns. Then add and subtract as normal but remember to write the
+ 0.85
decimal point in the answer.
O Find the difference.
4.12
-1.68
David goes for walks in a park every Sunday. He walks 2.62 km in the
morning and 1.3 km in the afternoon. How many kilometres does David walk
altogether every Sunday?
2.62 + 1.3 = ?
-ft
1
Write the numbers in a Fill in zeros in the empty Add or subtract
place-value table. Make spaces in the place- using column
sure that each digit is in value table so that the addition or
the correct place-value decimals you are adding subtraction.
column and that the or subtracting have the
decimal points are in line. same number of digits.

U . t h U . t h U . t h
2 .62 2 . 62 2 . 62
+ 1 .3 + 1 . 3 0—write zero + 1 . 30
9 2
t
line up the decimal points David walks 3.92 km
altogether every Sunday.

O Bo signed up for 3.3 gigabytes of mobile data. She has used 2.62 gigabytes.
How many gigabytes of data did she have left?
Regroup 3 units into 2 units 10
tenths. Add the tenths.
10 tenths + 2 tenths = 12 tenths
Regroup 3 tenths into 2 tenths
1
f and 10 hundredths.
U . t h U . t h U . t h
3 . 3 0-«—write zero 1^°
3 . 3
-2 . 62 +2 . 62 +2 . 6 2
0 . 68
t
line up the decimal points Bo had 0.68 gigabytes
of data left.

Check My
Understanding


Add.
a) 1 kg + 2.3 kg b) 34 g + 4.9 g c) 1g + 99.9g d) 4.53 g +1.22 9
e) 5.32 kg+ 1.71 kg f) 0.5 kg+ 0.05 kg g) 3.2g + 4.32g h) 3.7 kg + 0.21 kg

Work out the amount of change,
a) $5.30-$5.00 b) $2.67-$1.00 c) $12.00-$11.20 d) $21.00-$0.99
e) $4.53-$1.22 f) $5.20-$4.32 g) $2.99-$1.17 h) $0.89-$0.07
Work out each of the following.

a) 12.01+2.2 b) 9.09 + 1.1 c) 6.548 + 0.101 d) 4.21 +2.003 e) 4.99 + 0.01
f) 12.01-2.2 g) 3.452-2.132 h) 4.203-1.01 i) 0.5-0.05 j) 5.01-1.99
k) 16:5-6.7 I) 1.33 + 13.3 m) 6.52-0.2 n) 6.3 - 2.03 o) 5.099 + 0.001
p) 32.8-12.02 q) 43.62-41.8 r) 15.23 + 16.1 s) 0.5-0.05 t) 5.01 + 1.99
o Calculate.
a) 2.65 + 1.23 + 5.21 b) 2.05 + 0.33 + 7.2 c) 1.68 + 0.03 + 6.471
d) 0.05 + 1.99 + 0.201 e) 13.21 +0.943 + 0.708 f) 4.1 + 1.706 + 0.98

Fill in the missing digits in the blank spaces to make each statement true.
a) .3 + 1.2 = 7.5 b) 9.5 + .1 =9.6 c) 2.. .6 = 4.8
d) .4-1.1 =2.3 e) 9.4-_ _.3 = 7.1 f) 6.. .4 = 3.2

O* Challenge! Fill in the blanks to make each statement true,
a) 8.4- .5 = 7.9 b) 7. 5+15.1 = 2. .16 _.3+1 .2_ =12.58
Joe has added the decimals 6.01 and 7.3 incorrectly.
6.01
Correct his calculation and explain how he can add the decimals correctly.
+ 7.3
67.4



You have already learnt that decimals are often used in measurements.
Now let us look at more adding and subtracting decimals In everyday life.

Example 2


Mimi bought 2.38 kg of peanuts and 1.62 kg of raisins.
How many kilograms of snacks did she buy altogether?

+1 +1
Remember to line up the 2 . 38
decimal points. + 1 .62
4 . 0 0

Mimi bought 4 kilograms of snacks altogether.

O Ava has a piece of green ribbon that is 2.04 m long and a piece of red ribbon
that is 230 cm long. How much longer is the red ribbon in metres?


2 1 Remember to convert
wv* to the same unit of
2 0
-2.04
measurement.
-l/i
0 .26
The red ribbon is 0.26 m longer than the green ribbon.








Speed Challenge


How many can you work out correctly in 5 minutes?
Ready, set, go!

o


1. 47.2-10.7 = 1. 86.52-7.39 = 1. 17.51-16.6 =
2. 34.1 -23.0 = 2. 24.66 + 22.75 = 2. 25.08+ 18.7 =
3. 43.7-38.4 = 3. 62.88 + 25.33 = 3. 77.16+18.6 =
4. 30.5 + 14.8 = 4. 93.64-9.09 = 4. 49.19 + 3.64 =
5. 44.24-7.4 =
5. 36.2-35.7 = 5. 55.41 +28.51 =
6. 94.7-8.4 = 6. 12.28 + 10.45 = 6. 28.29-25.92 =
7. 20.7-20.3 = 7. 65.08-15.48 = 7. 93.73-0.6 =
8. 74.1 +21.6 = 8. 74.17 + 9.48 = 8. 99.65-28.79 =
9. 73.8 + 40.9 = 9. 54.46 + 8.19 = 9. 36.42-4.2 =
10. 48.0-45.5 = 10. 57.18 + 27.35 = 10. 74.4 + 27.16 =

• Number of • Number of • Number of
questions answered questions answered questions answered
in 5 min = in 5 min = in 5 min =









Exchange books with your classmate for peer marking. Write the score for each challenge.
• Were you able to find the sums or differences faster?
• Did you get more accurate?

Check My
Understanding


O A bee colony produced 0.272 kg of honey. The beekeeper collected 91 g of the honey. How
many kilograms of honey remains?
© The mass of a baby elephant was 206.99 kg. After two years, her mass increased by 108.95 kg.
Find the mass of the elephant after two years.
© Leo had a piece of rope 35.15 m long. He cut the rope into two pieces. If the length of one piece
of rope was 13.59 m, what was the length of the other piece of rope?
O The length of each side of a regular polygon is 6.3 mm. Its perimeter is 31.5 mm.

a) Find the number of sides the polygon has.
b) Name the polygon.




4.2.2 Multiplying whole numbers and

decimals by 10, ICQ and 1000







Investigate!
PAIR WORK
An elephant eats 0.236 tonne of food per day. How many tonnes will
the elephant eat in 10 days? We need to multiply 0.236 by 10.
Remember to fill in any
Study the table to see what happens when we multiply a decimal
spaces to the left of the
number by 10.
decimal point with zeros.
This gives the correct
itiv/ r • II
place value to the rest of
X 10 0 • ^2 /3 x6
the digits. 2 • 3^ 6^

When we multiply 0.236 by 10, the 2 digits moved from the tenths
column to the units column.
Fill in the blanks.
a) When we multiply 0.236 by 10, the digit moved from the
hundredths column to the tenths column.
b) When we multiply 0.236 by 10, the 6 digit moved from the
column to the column.
So when we multiply a number by 10, each digit moves one/two/three*
places to the left/right* in the place-value table. This makes a number
greater/smaller*.

How many tonnes of food would the elephant eat in 100 days? Study
the table to see what happens when we multiply a decimal number by
100.


UNIT 4 I Decimals

6
X 100 0 ^2^^3 ^
2-^ —3^ • 6^
Nick and Jamie recorded the
distances they ran. Nick ran
Fill in the blanks.
10 times as far as Jamie. If
a) When we multiply 0.236 by ICQ, the digit moved from the Jamie ran 0.86 km, how far
tenths column to the tens column,
did Nick run?
b) When we multiply 0.236 by 100, the digit 6 moved from the
column to the column.
So when we multiply a number by 100, each digit moves one/two/
three* places to the left/right* in the place-value table. This makes a
number greater/smaller*.

How many tonnes of food would the elephant eat in 1000 days? Study
the table to see what happens when we multiply a decimal number by
1000.
PlJ^ :
X 1000 0 •, --iJ— -—6
2* • •

Fill in the blanks.
a) When we multiply 0.236 by 1000, the digit _ moved from the
hundredths column to the tens column,
b.) When we multiply 0.236 by 1000, the digit 6 moved from the
column to the column.
So when we multiply a number by 1000, each digit moves one/two/
three* places to the left/right* in the place-value table. This makes a
number greater/smaller*.

* Circle the correct choice.

Example 3


O One pen costs $8.75. Mira wants to buy 10 pens. How much will it cost?

Multiply $8.75 by 10. When you multiply by 10, each digit moves one
place to the left. This makes the number greater.
$8.75 X 10 = $87.50


O Dion wants to convert $12.04 to cents.

Multiply $12.04 by 100 When you multiply by 100, each digit moves two
places to the left. This makes the number greater.
$12.04 X 100 = 1204 cents


O A brick has a mass of 3.5 kg. What is the total mass of a load of 1000 bricks?

Multiply 3.5 kg by 1000 When you multiply by 1000, each digit moves three
3.5 kg X 1000 = 3500 kg places to the left. This makes the number greater.
o

Check My
Understanding



Multiply,
a) 3.2 X 10 b) 0.9 X 10 c) 1.6 X 100 d) 2.32 X 100
e) 0.04x10 f) 0.03 X 100 g) 3.04 X 10 h) 5.09 X 100
1) 15.3 X 10 j) 2.6X1000 k) 1000 X 0.02 I) 1000 X 1.04

Fill in the blanks.
a) 2.6 X = 26 b) 4.03 X _ = 403 c) 60.2 X = 602
d) 0.32 X = 32 e) 3.02 X _ = 3020 f) 0.14 X = 140
g) X 0.26 = 26 h) X 2.09 = 20.9
O A toy costs $23.99. How much would
a) 100 of the same toys cost?
b) 1000 of the same toys cost?
O Jo drinks 1.8 i of water every day. How many litres of water will he drink in 10 days?
O Use the numbers in the box to write as many different multiplication number sentences
as you can. Each number can be used more than once.
Example: 0.3 x 10 = 3

a)
0.3 30 300 3
10 100 1000 0.6
0.6 0.1 0.003



b)
0.07 0.7 7 70 700
10 100 1000 10000



Work with a partner. Members of the owl family have different opinions about
multiplying a decimal number by 10. Analyse each owl's statement and give feedback as
to why you think each statement is correct or incorrect.


When you multiply
answer
by 10, you add a zero. What do you
0.8 X 10 = 0.80 think?





0.8 X 10




You need to move each
You need to move each
digit two places to the left. digit one place to the left
0.8 X 10 = 80 0.8 X 10 = 8.0


Decima s

4.2.3 Multiplying decimals In this chapter
Pupils should be able to:
• multiply decimals with
Example 4
one and/or two places
by single-digit numbers
O Kifn uses 13.7 i of water to wash her car. How many litres of water will she
need to wash 8 identical cars?

We need to find the product of 13.7 x 8. ^ RECALL
+2 +5
There is one digit after the decimal point.
13.7^ Find the product.
X 8
To place the decimal point in the 15.8
109. 6-^-
correct position in the answer, count X 4
KJ
one digit from the right in the answer.
She will need 109.6 litres of water to wash 8 identical cars.

O Mia needs 2.12 metres of material to make one curtain. How many metres of
material will Mia need to make 3 identical curtains. We need to multiply.
2.12 X 3 = ?
There are two digits after the decimal point.
2.12'
X 3
To place the decimal point in the correct
6.36 '
position in the answer, count two digits
from the right in the answer.

Mia will need 6.36 m of material to make 3 identical curtains.


Check My

Understanding


Multiply.
O a) 1 X 2.3 b) 3 X 4.2 c) 0.5 X 2 d) 0.1 X 0 e) 2 X 7.8

f) 4 X 1.9 g) 5.3 X 3 h) 0.9 X 9 I) 4.4 X 4 j) 9 X 8.9
a)
0 1 X 2.38 b) 2 X 4.32 c) 0.51 X 4 d) 0.07 X 3 e) 2 X 7.88
f) 6X1.29 g) 5.03 X 6 h) 0.88 X 6 i) 4.06 X 4 j) 9 X 17.9
a)
0 0.8 X 8 b) 5 X 4.02 c) 6.5 X 6 d) 1.2 X 9 e) 2 X 0.32
f) 9.09 X 2 g) 4.5 X 7 h) 18.23 X 5 i) 23.07 X 3 j) 52.5 X 2
O Fill in the missing numbers.

a) X 1.2 = 2.4 b) 3.1 X = 9.3 c) 2.2 X = 8.8
d) X 6 = 6.66 e) X 4= 12.8 f) X 5 = 12.25

0 Sue says that 2.34 X 2 = 4.68.
Chris says that 2.34 x 2 = 46.8.
Who is correct? Give a reason for your answer.

In this chapter 4.2.4 Divide whole numbers and
Pupils should be able to:
decimals by 10,100 and 1000
• divide whole numbers
and decimals by 10,100
and 1000
Investigate


Jane is training for a cycling race. She needs to cycle 1832 km in
10 weeks. How many kilometres should she cycle per week (assuming
she cycles the same distance each week)?
To calculate this, Jane needs to divide 1832 by 10. Look at
the table to see what happens when we divide a whole number by 10.



v 10 1\ 8\ 3\ 2~~~. •
^8 ^3 •

Fill In the blanks.
a) When we divide 1832 by 10, the digit moved from the hundreds
column to the tens column,
b) When we divide 1832 by 10, the digit 2 moved from the
column to the column.

So when we divide a number by 10, each digit moves one/two/three*
places to the left/right* in the place-value table. This makes a number
greater/smaller*.

If Jane wants to cycle 1832 km in 100 weeks, how many kilometres
should she cycle per day (assuming she cycles the same distance each
day)?
Look at the table to see what happens when we divide a whole number
by 100. ^1




o
o
1
3
2
Fill in the blanks, • ""8
a) When we divide 1832 by 100, the digit _ moved from the
hundreds column to the units column.
b) When we divide 1832 by 100, the digit 2 moved from the
column to the column.
So when we divide a number by 100, each digit moves one/two/three*
places to the left/right* in the place-value table. This makes a number
greater/smaller*.







Decimals

O Look at the table to see what happens when we divide a whole number
o
by 1000. Spotlight

1 ^ _ m
Addition and subtraction
1_ 8-^ _3-- 2 — •
-rIOOO
are inverse operations.
-Tl i"— -^2
Multiplication and
Fill in the blanks. division are inverse
operations.
a) When we divided 1832 by 1000, the digit moved from the
thousands column to the units column. Inverse operations are
b) When we divided 1832 by 1000, the digit 8 moved from the operations that undo
column to the column. each other.
Addition Subtraction
So when we divide a number by 1000, each digit moves one/two/
three* places to the left/right* in the place-value table. This makes a
number greater/smaller*.
Multiplication Division
*Circle the correct choice



Example 5


With measurements, we often work with decimals.
O Kyle has 432.1 m of rope. He needs to cut the rope into 10 equal pieces. How
long should each piece be? Give your answer in metres.

Divide 432.1 by 10. When you divide by 10, each digit moves one
place to the right. This makes the number smaller.
432.1 -r 10 = 43.21 metres



O Harry has 9192.3 kg of fish. He needs to serve 100 guests. How many grams of
fish are there in each serving?
Divide 9192.3 by 100. When you divide by 100, each digit moves two
9192.3 ^ 100 = 91.923 g places to the right. This makes the number smaller.




O Gale has run 3561.2 m. How many kilometres did Gale run?
Divide 3561.2 by 1000. When you divide by 1000, each digit moves
three places to the right. This makes the
3561.2-j-1000 = 3.5612 km
number smaller.














©

Check My

Understanding


Divide.
a) 32-MO b) 0.9-r 10 c) 160-f 100 d) 2.32^ 100
e) 40v10 f) 300-r 100 g) 3.04 V 10 h) 50.9^100
i) 15.3-r 10 j) 265.1 ^1000 k) 103.14-r 1000 I) 3.04^100

Fill in the blanks.
a) 26-r = 2.6 b) 14.3-r _ = 0.143 0 60.2^ _ = 60.2 d) 324 -r _ = 0.324
e) 605-r = 60.5 f) 0.14-r. = 0.014 9) 7.2- = 0.072 h) 1000 -r = 10
e Jo drinks 21 t of water in 10 days. How many litres of water does he drink each day if he drinks the
same amount every day?
o A cellphone is on sale at one tenth of its original price. If the original price of the cellphone was
$289.98, what is the sale price? Give your answer to the nearest dollar.



Inthischap,er Dividing decimals


Pupils should be able to:
• divide decimals with one
Example 6
and/or two places by
single-digit numbers
Sasha has 6.4 i of juice. She wants to share it equally between her 2 children.
How many litres of juice does each child get?
^ RECALL Divide 6.4 by 2.
To place the decimal point in the correct position in the
Find the quotient. answer, count one digit from the right in the answer.
3
2r6 4 There is one digit after the decimal point.
I
9 43.2
Each child gets 3.21 of juice.
Ben has 4.35 gigabytes of data. If Ben uses all the data in 5 days and that he
uses the same amount of data each day, how much data does he use per day?

Divide 4.35 by 5.
o Write zero to show that 4
Spotlight
cannot be divided by 5.
What is 0 5? There is I To place the decimal point in the correct position in the
nothing to be shared
0 . 8 7 < answer, count two digits from the right in the answer.
equally into 5 groups,
so zero divided by any 5|4 3 5 There are two digits after the decimal point.
4 0
number is zero.
3 5
What is 5 0? It is not 3 5
possible to divide a
number by zero. So it is
He uses 0.87 gigabytes of data per day.
undefined.

0 Nick has 2.6 kg of sugar. He wants to pack the sugar into 4 packets of the
same mass. How many grams of sugar should Nick pack in each packet?
Divide 2.6 by 4.

0. 65
2 . 6 0- This zero is added to help you divide.
2 . 4
2 0
Remember 1 kg = 1000 g
2 0
Convert to the required
0.65 kg X 1000 = 650 g unit of measurement.
Nick should pack 650 g of sugar in each packet.

Check My

Understanding


O Divide.
a) 4.3-fl b) 6.3^3 c) 4.8-5-4 d) 5.5-5-5 e) 14.2^2
f) 0.6-r 2 g) 2.5^5 h) 0.9 -5- 9 i) 3.6 4 j) 1.4^2
0 Divide.
a) 2.34-rl b) 2.32^2 c) 6.06-5-6 d) 0.27 -r 3 e) 5.35 -r 5
f) 6.21 -^3 g) 6.06^4 h) 0.54-f 6 i) 9.36 V 9 j) 4.32 -r 8
0 Divide.
a) 0.8-r 8 b) 45.9-r 5 c) 0.36-r 6 d) 14.35 V 7 e) 9.09 ^ 2
f) 9.09-r 3 g) 49.14-^7 h) 25.75 -5- 5 I) 4.02^5 j) 18.23-5-5
O Fill in the missing numbers.
a) -^4=1.22 b) 16.4-5- = 4.1

c) 2.24-r =1.12 d) -5-3 = 3.1
e) -r4 = 3.2 f) -5- 6 = 4.4
0 Each sheet of metal is 1.25 cm thick. If Mei stacks 7 sheets on top of each other, how thick will the
stack be?
0 Brenda has 3 travel bags. The total mass of all the bags is 9.69 kg. What is the mass of each bag if
each bag has the same mass?
0 The price of 1 kg of white rice is S$3.39. How much will
5 kg of rice cost?
A
0 factory used 60.64 kg of noodles to make 4 equal batches of noodle soup. How many
grams of noodles did the factory put in each batch of soup?
A
0 factory produces 0.5 m of copper wire every minute. How many metres of wire will
the factory produce in
a) 6 minutes?
b) 1 hour?
Jamie has 0.46 m of rope. If he cuts the rope in half, what is the length of each piece of rope?
Sam says that 4.14 -r 2 = 2.7.
Cleo says that 4.14 -2 = 2.07.
Who is correct? Give a reason for your answer.
o

CHAPTER 4.3
In this chapter

Pupils should be able to:
Tdering Decimals
• order decimals including
measurements,
in ascending and
descending order Scientists measure the magnitude
(strength) of earthquakes using
• change measurements
the Richter scale. The Richter scale
to the same units before
measures the magnitude of an
ordering them
earthquake on a scale of 1 to 10. The
Richter scale measures earthquakes
in decimal numbers to tenths. Most
earthquakes register 2.5 or less and
Which earthquake was the are too small to be noticed by people.
strongest? Earthquakes with a magnitude of 8.0
Look at the magnitude or higher on the Richter scale occur
of the following three about once every 5 to 10 years.
earthquakes. Work with
your partner to name the
4.3.1 Comparing and ordering
earthquakes in order from
the strongest (highest
decimals
magnitude) to the weakest
(lowest magnitude).
When you order any numbers, including decimals, you place them in order from:
• The Tohoku
earthquake (Japan
2011) had a Greatest to smallest Smallest to greatest
magnitude of 9.1 on (this is descending (this is ascending
the Richter Scale. order) order).
• TheValdivia
earthquake (Chile
1960) had a
Example 7
magnitude of 9.5 on
the Richter Scale.
Write these numbers in ascending order:
• The Prince William
Sound earthquake 1
6.45 6 4 5 Place-value tables are
(Alaska 1964) had a •
helpful when ordering
5.52 5
magnitude of 9.2 on • 5 2
decimals.
6.48
the Richter Scale. 6 • 4 S
5.68 5 • 6 8
In this example, all decimals begin with a digit in the units place.
In these decimals, the units have the highest place value.
You compare the units first.

The smallest value in the units place is 5.
5 <6.



UNIT 4 I Decimals

There are two decimals that have the digit 5 in the units place, so look at the
tenths place. Compare the digits in the tenths place:
5 tenths < 6 tenths.
5.52 is smaller than 5.68.
5.52 < 5.68

Now let us look at the decimals that have 6 in the units place.
They have the same value in the tenths column, so look at the hundredths place.
Compare the digit in the hundredths place:
5 hundredths < 6 hundredths.
6.45 < 6.48.
So, the decimals arranged in an ascending order are:

5.52 5.6 6.45 6.48
ascending









GROUP WORK
Get into groups of four.
.
...iSM't S w- ..l-- , • . .
O There are four sets of decimal cards with four decimals in each set. Each 3.7 3.203 3.434 3.443
group will choose one set of decimal cards as shown on the left.
O Give each group member one decimal card. Hold the card in front of
you.
O Without talking to each other, members in each group must arrange
themselves in descending order by standing in a row.
O The first group to stand in order neatly must raise their hand to get the
teacher's attention.
@ The fastest group wins!





























Q

Check My

Understanding


O Arrange the volumes in order.
a) 4.32U 1.203^ 0.38 £ 2.741 i
From smallest to greatest
b) 1.729 £ 1.279^ 1.20U 1.9i
From greatest to smallest
c) 4.29U 4.192 i 4.102« 4.2 i
From smallest to greatest
d) 10.31 £ 10.13 £ 13030 m^ 3030 m^
From smallest to greatest
O Arrange the following masses in ascending order.

a) 0.6 kg 0.25 kg 0.9 kg 0.1 kg
b) 4t 0.81 3.611 3.61
c) 2.43 g 2.5 g 2.7 g 1.066 g
d) Og 10.1 g 10.87 g 10.78 g
e) 2.07 kg 2.70 kg 2.71 kg 2.172 kg

O Arrange the following distances from longest to shortest.
a) 1.35 km 1.6 km 1 km 0.16 km
b) 2.2 m 2.5 m 2.51 m 2.01 m
c) 0.08 km 0.081 km 0.18 km 0.108 km
d) 2.23 mm 2.32 mm 20.3 mm 20.32 mm
e) 6.51 cm 6.15 cm 60.1 mm 6.05 cm
O Challenge! Using each group of digits to form the smallest possible
decimal.
• Use all the digits in each group
• Include a decimal point
• Use each digit only once
• Form positive decimals only

1 9 0 2 7
2 5 7 3 9
1 7 0 1 3
8 0 0 0 2

2 2 0 4 0
Q*Challenge! Kim has the following cards.








Her teacher asks her to arrange all the cards on the board to make a
decimal as close as possible to the number 6. Show Kim how to arrange
her cards.



Decimals

4.32 Ordering decimals with different

units of measurement


Sometimes it is helpful to compare two or more quantities such as travelling time, E
income, expenses, size, mass, speed, temperature, height and distance. It is easier
t RECALL
to compare and order decimals that have the same unit of measurement. For
£
example, it is difficult to compare a length in kilometres with a length in metres.
We need to change both quantities to have the same unit before comparing. uengin ivies
Conversion Operation
Example 8 mm m -r 1000

m —*■ mm X 1000
O Arrange the masses from heaviest to lightest.
cm ^ m ■rlOO
0.5 kg 560 g 1.25 kg 1020 g
m —cm X 100
m —> km ■r 1000
Convert all masses to the same unit. In this example,
1 we will convert kg to g. X 1000
1 ...
0.5 kg X 1000 = 500 g
Recall that 1000 g = 1 kg. Operation
1.25 kg X 1000 = 1250 g Conversion
g-^kg 1000
Order the masses from heaviest to lightest. kg-^g X 1000
^ 1250g 1020g 560g 500 g
Conversion Operation
Write down the masses in order from heaviest to lightest using the
m£^- £ ■r 1000
original units given.
ml X 1000
1.25 kg 1020 g 560 g 0.50 kg


O Write the distances in order from shortest to longest.

2.5 m 273 cm 2.05 m 256 cm
Convert the distances to the same unit. In this example, we will
convert cm to m.
273 cm 100 = 2.73 m

256 cm 100 = 2.56 m
Check My
Order the lengths from shortest to longest.
Understanding
2.05 m 2.5 m 2.56 m 2.73 m
Convert these measurements,
Write down the lengths in order from shortest to longest using the
original units given. a) 28.5 cm = m

2.05 m 2.5 m 256 cm 273 cm
b) 1.706 m = cm







o

Check My
Understanding


O Arrange the measurements in descending order.

a) 250 g 0.55 kg 0.525 kg 1250 g
b) 250 m^ 2.5^ 250^ 25 m^
c) 325 m 3.25 km 3.251 km 3251 cm
d) 302 mm 0.3 m 320 mm 0.03 m

e) 101 cm 0.99 m 909 mm 98 cm
During an experiment, Cheng measured the growth of green beans under different
conditions. He recorded the heights of the bean plants 18 days after planting the seeds.

Bean Height of Bean Plant
A 1.01 cm
B 11.4 mm
C 1.11 cm
D 11.04 mm

Cheng wants to compare the heights of each bean plant to find
out which bean grew the fastest. Help him order the heights of
the bean plants in descending order.



The reading on Stopwatch A is 3 minutes and 52.17 seconds.
Write the following times in order from fastest to slowest.
Stopwatch A Stopwatch B Stopwatch C Stopwatch D
✓-X <-X


5:50.0^ d: 1.50
1


3 X /■ X 3 X ^ X

o Goh is writing an article in the local newspaper comparing the
prices of the silver pomfret fish at a fish market. He compared
the prices of the fish from four different fish shops.

Shop Price per kg
A $20.35

B $29.99
C $20.89
D $29.90

Arrange the shops in order of price from cheapest to most
expensive.


UNIT 4 I Decimals

CHAPTER 4.4

r •vw ' -vsma

In this chapter
Rounding Numbers Pupils should be able to:
• round whole numbers
to the nearest 10, 100
The attendance at the Singapore Grand Prix in 2018 was not exactly 260 400 or 1000 and decimals,
people. When working with numbers, we do not always need to be exact. including measurements,
Rounding is a way of simplifying numbers to make it easier to describe and to the nearest whole
number or one decimal
understand the numbers.
place
blivUDU' iijf
II lUhl UK!
RTTCiMnn!\!.
I I C1 (U M K i_ L

'RLIHES





















^ RECALL

Rounding numbers to the O Round each number

to the nearest
nearest whole number or
hundred,
1 decimal place a) 564 b)1205
e Round each number
We round numbers to give an approximate value that is close to the real value. to the nearest tenth,
Rounding of numbers can make calculations easier. a) 4.51 b) 30.99

You can round numbers to the nearest whole number, 10, 100, 1000 or other
multiples of 10.

The distance between Singapore and Beijing is 4484.048 km.
• If you round the distance to the nearest whole number, it is approximately
In the example on the
4484 km.
left, which estimated
• If you round the distance to the nearest 10, it is approximately 4480 km.
value here is closest to
• If you round the distance to the nearest 100, it is approximately 4500 km.
the actual value? Explain.
You can round numbers to a given number of decimal places.


o

Pi (ji) is a special number
3.141592653589793238462
that is used in calculations
involving circles. It is obtained 6433832795028841971693
by dividing the circumference
9937510582097494459230
of a circle by its diameter. The
7816406286208998628034
answer is a decimal. Pi is a
decimal with a never ending 8253421170679821480865
number of decimal places.
To simplify working with
the value of pi, it is usually
rounded off to two decimal
places and written as 3.14. s Figure 4.1 Pi has a never ending number of
decimal places.

0i Investigate!


Look at the number line. Can you see a pattern? When do we round up to
the bigger whole number? When do we round down to the smaller whole
number?

O Fill in the missing numbers. The first one has been done for you.
19
a) 19.1 is rounded to | b) 19.2 Is rounded to
c) 19.3 is rounded to | d) 19.4 is rounded to

e) 19.5 is rounded to | f) 19.6 Is rounded to

g) 19.7 is rounded to h) 19.8 Is rounded to

i) 19.9 is rounded to
0 Use your answers In Question 1 to write down some conclusions about

rounding decimals to the nearest whole number.
a) When the digit in the tenths column is 1, 2, 3 or 4, then the number
is rounded to the whole number .
b) When the digit in the tenths column Is 5, 6, 7, 8 or 9, then the
number Is rounded to the whole number .





19 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20


When we a number to the nearest whole number, we look
at the digit in the place.
If the digit < 5, we round It to the _ whole number,
If the digit = 5, we round It to the _ whole number,
If the digit > 5, we round It to the _ whole number.





UNIT 4 I Decimals

Investigate rounding decimals between 7.1 and 7.2 to one decimal place.

\
7.1 and 7.2 are examples
1 • 1
of numbers with one
7 • 2 decimal place.

Observe the pattern in rounding decimals to the nearest one decimal place.
Use a number line to help you.







7.1 7.11 7.12 7.13 7.14 7.15 7.17 7.18 7.19 7.2

O Fill in the missing numbers. The first one has been done for you.

a) 7.11 is rounded to 7.1 b) 7.12 is rounded to

c) 7.13 is rounded to d) 7.14 is rounded to

e) 7.15 is rounded to f) 7.16 is rounded to
g) 7.17 is rounded to h) 7.18 is rounded to

i) 7.19 is rounded to

O Use your answers in Question 1 to write down your conclusions for
rounding to one decimal place.
a) When the digit in the hundredths column is 1, 2, 3 or 4, then the
number Is rounded to the decimal .
b) When the digit in the hundredths column is 5, 6, 7, 8 or 9, then the
number is rounded to the decimal .
When we a number to the nearest 1 decimal place, we look
at the digit in the place.
If the digit < 5, we round it to the tenth,
If the digit = 5, we round it to the tenth,
If the digit > 5, we round it to the tenth.




















A

O Garth used 4.25 g of cheese to make a cheeseburger. Round the mass of cheese used to
one decimal place.
Find the digit in the place-value column you are rounding to. Underline this digit.
1
4.25
Look at the digit to the right of the underlined digit.
4.25
If the digit to the right is 5 or more, add 1 to the digit in the place-value column
you are rounding to. Drop all the digits to the right of the place-value column you
are rounding to.
cm
The digit is 5. So we add 1 to 5 and drop all digits to the right to round up.
4.25 g rounded to one decimal place is 4.3 grams.
O Garth used 4.21 g of cheese to make another cheese burger the next day.
Round the mass of cheese used to one decimal place.

Find the digit in the place-value column you are rounding to. Underline this digit.
4.21

Look at the digit to the right of the underlined digit:
4.21

If the digit to the right is 5 or more, add 1 to the digit in the place-value column
you are rounding to. Drop all digits to the right of the place-value column you are
rounding to.

The digit is 1 which is less than 5.
If the digit to the right is less than 5, add nothing to the place you
are rounding to. Drop all the digits to the right of the place you are rounding to.
The digit is 1, so we drop all digits to the right and round down.

4.21 g rounded to one decimal place is 4.2 g.



Check My

Understanding

O Complete the table.


Round to nearest
Decimal 10 ICQ 1000 whole number one decimal place

a) 1432.3
b) 645.5
c) 937.06
d) 787.19
e) 7847.9
f) 1009.99


Decimals