Word Problems Leading to Simple Equations

Word problems give us a first glimpse into how mathematics is used in the real world. To be solved, a word problem must be translated into the language of mathematics, where we use symbols for numbers – known or unknown, and for mathematical operations.

Example 6.4.1: 

Think of a number, add 5, the result is 10. What is the number? 

Solution 

Let the number be y

Add 5 to y → \( \scriptsize 5 \: + \: y \)

Result → \( \scriptsize 5 \: + \: y = 10 \)

\( \scriptsize 5 \: + \: y = 10 \)

Collect like terms

\( \scriptsize y = 10 \: – \: 5 \)

\( \scriptsize y = 5\)

∴ the number is 5 

Example 6.4.2:

The smallest of three consecutive odd numbers is n. if their sum is 27, find the three numbers. 

Solution 

Odd numbers = 1, 3, 5, 7, 9…

Difference between each consecutive odd number = 3 – 1 = 5 – 3 = 7 – 5 = 9 – 7 = 2 

⇒ let the number be n, n + 2, n + 2 + 2 

i.e. n, n + 2, n + 4

From the question, their sum is 27

n + n + 2 + n + 4 = 27

3n + 6 = 27 

Collect like terms 

3n = 27 – 6 

3n = 21 

Divide both sides by 3 

\( \frac{3n}{3} = \frac{21}{3} \)

\( \scriptsize n = 7 \)

Let us substitute the value of n back into the equation

n, n + 2, n + 4

= 7, 7 + 2, 7 + 4

The smallest number = 7 

Next number = 7 + 2 = 9 

Next number = 7 + 4 = 11

⇒ The numbers are 7, 9 and 11

Note: 
For consecutive odd numbers, use: x, x + 2, x + 4, x + 6, x + 8, x + 10, …
For consecutive even numbers, use: x, x + 2, x + 4, x + 6, x + 8, …
For consecutive numbers, use: x, x + 1, x + 2, x + 3, x + 4, …

Example 6.4.3:

The age of a boy is twice that of his sister. Five years ago, the boy was 3 times the age of his sister. How old are the boy and the girl now? 

Solution: 

You have to know the sister’s age before the boy’s age. So let the sister’s age be x

The age of a boy is twice that of his sister = \( \scriptsize 2 \: \times \: x = 2x\)

5 years ago: 

Sister’s age = \( \scriptsize x \: – \: 5\)

Boy’s age = \( \scriptsize 2x \: – \: 5\)

Five years ago, the boy was 3 times the age of his sister.

\( \scriptsize 2x \: – \: 5 = 3(x \: – \: 5) \)

\( \scriptsize 2x \: – \: 5 = 3(x) \: +3 (-5) \)

\( \scriptsize 2x \: – \: 5 = 3x \: – \: 15 \)

Collect like terms 

\( \scriptsize 2x \: – \: 3x = -15 \: + \: 5 \)

\( \scriptsize -x = -10 \)

\( \scriptsize x = 10 \)

⇒ The girl’s age now = 10 years 

The boy’s age now = 2 × 10 = 20 years 

Example 6.4.4: 

Think of a number, divide it by 5 and then subtract 4. The result is 15. Find the number.

Solution 

Let the number be y 

Divide y by 5 → \( \frac{y}{5} \)

Subtract 4 → \( \frac{y}{5} \scriptsize \: – \: 4 \)

Result → \( \frac{y}{5} \scriptsize \: – \: 4 = 15 \)

Solve 

\( \frac{y}{5} \scriptsize \: – \: 4 = 15 \)

Multiply through by 5 

\( \frac{y}{5} \scriptsize \: \times \: 5 \: – \: 4 \: \times \: 5 = 15 \: \times \: 5 \)

\( \frac{y}{\not{5}} \scriptsize \: \times \: \not{\!5} \: – \: 4 \: \times \: 5 = 15 \: \times \: 5 \)

\( \scriptsize y \: – \: 20 = 75 \)

Collect like terms 

\( \scriptsize y = 75\: + \: 20 \)

\( \scriptsize y = 95 \)

∴ The number is 95

Example 6.4.5:

Fatima is x years old and her brother is half of her age. The sum of their ages is half of their father’s age who is now 60. How old are Fatima and her brother? 

Solution 

Fatima’s age = x

Brother’s age = \( \frac{1}{2} \scriptsize \: of \: x \\ = \frac{1}{2} \scriptsize \: \times \: x \\ = \frac{x}{2} \)

Sum of Fatima’s age and brother’s age = \(\scriptsize x \: + \: \normalsize \frac{x}{2} \\ = \frac{2x \: + \: x}{2} \\ = \frac{3x}{2} \)

Sum of Fatima and brother’s age = \( \frac{3x}{2} \)

Father’s age = 60

Half of father’s age = \( \frac{1}{2} \scriptsize \: \times \: 60 = 30 \)

Sum of Fatima and brother’s age = \( \frac{1}{2} \scriptsize \: of \: father’s \: age \)

\( \frac{3x}{2} \scriptsize = 30 \)

Multiply both sides by 2

\( \frac{3x}{2} \scriptsize \: \times \: 2 = 30 \: \times \: 2 \)

\( \scriptsize 3x = 30 \: \times \: 2 \)

\( \scriptsize 3x = 60\)

Divide both sides by 3 

\( \frac{3x}{3} = \frac{60}{3} \)

\( \frac{\not{3}x}{\not{3}} = \frac{60}{3} \)

\( \scriptsize x = 20\)

Fatima’s age = x = 20years

Fatima is 20 years old and her brother is \( \frac{20}{2} \scriptsize = 10 \: years \: old \)

Evaluation Questions: