Translation of Word Problems into Equations

To translate a word problem into an algebraic equation and solve it, the following are the steps:

1. Read the questions carefully and then decide what the unknown number is.
2. Where necessary, change all the units of measurement to the same unit.
3. Use a letter to represent the unknown
4. Use the information provided to write the required equation.
5. Solve the equation as usual.
6. Use the solution obtained to answer the questions in words.
7. You can check your answer as usual.

Example 5.3.1:

The age of a father is thrice the sum of the ages of his two sons. After 5 years his age will be twice the sum of their ages. Find the present age of the father.

Solution

Step 1: Let us understand the given information. There are two sets of information given in the question.

a) The age of the father is thrice the sum of the ages of his two sons (At present).
b) After 5 years, his age would be twice the sum of their ages (After 5 years).

Step 2: Target of the question = Present age of the father.

Step 3: Introduce the required variable for the information given in the question.

Let x be the present age of the father

Let y be the sum of the present ages of two sons.

Clearly, the value of x is to be found because that is the target of the question.

Step 4:

Translate the given information as a mathematical equation using x and y.

First Information: The age of the father is thrice the sum of the ages of his two sons.

Translation i: The age of the father =  x

x  =  thrice the sum of the ages of his two sons = 3y

Equation related to the first information using x and y is   

x  =   3y  ___________ (i)

Second Information: After 5 years, his age would be twice the sum of their ages.

Translation (ii): Age of his father after 5 years = x  +  5

Sum of the ages of his two sons after 5 years  =  y  +  5  +  5 =   y + 10

(Here we have added 5 two times. The reason is there are two sons).

Twice the sum of ages of two sons = 2(y  +  10)

Equation related to the second information using x and y  is   

x  +  5   =  2 (y  +  10)  _____________  (ii)

Let’s list out our two equations

x  =   3y  ___________ (i)
x  +  5   =  2 (y  +  10)  _____________  (ii)

Step 5: Solve equations (i) and (ii)

Substitute  x   =  3y  from equation (i) into equation (ii), i.e “x  +  5   =  2 (y  +  10)”

We now have;  

3y  +  5   =  2(y  +  10)

open the brackets

⇒ 3y  +  5   =  2y  +  20

collect like terms

⇒ 3y  –  2y   =  20  – 5

y   =   15

Substitute y  =  15 into equation (i)   “x  =   3y“

x  =   3 (15)

x  =   45 

Therefore, the present age of the father is 45 years.

Example 5.3.2:

Translate the following statements into algebraic equations and solve them.

i. Think of a number, add 6, and the result is 10. What is the number?
ii. A number is added to 8 and the result is multiplied by 5 and then 10 is added. If the final answer is 35. What is the number?

Solution

i. Think of a number ____ let the number be x

Add 6 ________  x  +  6

The result  ______ x  +  6   =  10

Then solve the equation   x  +  6   =   10

Subtract 6 from both sides

x  +   6  –  6   =  10   –  6

x  =  4

ii. A number is added to 8  ____ let the number be x

∴ x + 8

and the result is multiplied by 5 ____  5(x  +  8)

and then 10 is added ____ 5(x  +  8)   +  10

If the final answer is 35 ____ 5(x  +  8) + 10    =  35

What is the number?  ____    Solve, 5(x  +  8)  +  10   =  35

5(x  +  8)  +  10   =  35

Open the brackets

5x   +  40   +  10   =   35

5x  +  50    =  35

Collect like terms 

5x   =  35  –  50

5x  =  -15

Divide both sides by 5

⇒ \( \frac{5x}{5} = \frac{-15}{5} \\ \frac{\not{5}x}{\not{5}} = \frac{-15}{5} \\ \scriptsize x = \; -\; 3\)

The number is  -3