Using letters to represent numbers
🎯 In this topic you will
- Find the value of a letter that represents a number.
- Use the idea that an unknown can represent a variable rather than a single fixed number.
🧠 Key Words
- constant
- variable
Show Definitions
- constant: A value that stays the same and does not change.
- variable: A quantity represented by a letter that can take different values.
🔍 Using Letters to Represent Numbers
In this unit, you will use a letter to stand for a number as in these examples. Letters such as $a$ and $b$ can represent unknown values that we can work out using the information given.
❓ EXERCISES
1. Cheng plays a board game using a dice. He uses the instructions together with his dice score to work out how many spaces he moves. $d$ represents the dice score.
Work out how many spaces Cheng moves.
a. $d - 3$
b. $6 - d$
c. $4 + d$
👀 Show answer
a. $5-3=2$ spaces
b. $6-5=1$ space
c. $4+5=9$ spaces
2. For each pair of expressions write ‘equal’ or ‘not equal’.
a. $d+4$ $4+d$
b. $2+d$ $d+2$
c. $5-d$ $d-5$
👀 Show answer
b. equal
c. not equal
3. Khalid says, ‘$x+3$ is the same as $3+x$ so $x-3$ must be the same as $3-x$.’ Is Khalid correct? Explain your answer.
👀 Show answer
4. Martha buys $2$ more pairs of socks than shoes.
a. Copy and complete the table where $x$ represents the number of pairs of shoes and $y$ represents the number of pairs of socks.
| $x$ | $1$ | $2$ | $?$ |
|---|---|---|---|
| $y$ | $3$ | $4$ | $6$ |
b. Write a number sentence linking $x$, $y$ and $2$.
👀 Show answer
b. $y=x+2$.
5. This puzzle has $9$ pieces. Hassan places $1$ or more shapes and Sanjay places the other shapes.
a. Copy and complete this table to show the number of pieces each person places.
b. Write a number sentence linking $x$, $y$ and $9$.
👀 Show answer
b. $x+y=9$.
6. $a$ and $b$ represent the lengths of two strips of card. $b$ is $3$ cm longer than $a$. The two strips are placed end to end. The total length is $15$ cm. Find the lengths of $a$ and $b$.
👀 Show answer
So $a+(a+3)=15 \Rightarrow 2a+3=15 \Rightarrow a=6$ and $b=9$.
7. The perimeter ($p$) of a square is the sum of the lengths of the sides. $s$ represents the length of a side measured in centimetres. $p=s+s+s+s$.
a. What is the value of $p$ when $s=5$?
b. What is the value of $p$ when $s=7$?
c. What is the value of $s$ when $p=32$?
👀 Show answer
b. $p=4\times7=28$
c. $4s=32 \Rightarrow s=8$
8. This isosceles triangle has a perimeter of $15$ cm.
a. Find three possible sets of values for $x$ and $y$.
b. Write a formula for the perimeter ($p$) of the triangle using $x$ and $y$.
👀 Show answer
$x=5,\ y=5$
$x=6,\ y=3$
$x=4,\ y=7$
b. $p=2x+y$.
🧠 Think like a Mathematician
$a$, $b$ and $c$ each represent a whole number from $1$ upwards.
$a + b + c = 7$
Find all the possible values for $a$, $b$ and $c$. How many different solutions can you find?
Show Answers
Since $a,b,c \ge 1$ and $a+b+c=7$, list all positive integer triples:
- $(1,1,5)$
- $(1,2,4)$
- $(1,3,3)$
- $(1,4,2)$
- $(1,5,1)$
- $(2,1,4)$
- $(2,2,3)$
- $(2,3,2)$
- $(2,4,1)$
- $(3,1,3)$
- $(3,2,2)$
- $(3,3,1)$
- $(4,1,2)$
- $(4,2,1)$
- $(5,1,1)$
Total number of solutions:$15$.