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 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 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 xÂ andÂ yÂ isÂ Â Â

xÂ +Â 5 Â =Â 2 (yÂ +Â 10)Â _____________Â **(ii)**

Let’s list out our two equations

xÂ = Â 3yÂ ___________ **(i)**

xÂ +Â 5 Â =Â 2 (yÂ +Â 10)Â _____________Â **(ii)**

**Step 5: **Solve equationsÂ **(i)** and **(ii)**

SubstituteÂ xÂ Â =Â 3yÂ from equation **(i)** into equation **(ii)**, i.e “**x**Â +Â 5 Â =Â 2 (yÂ +Â 10)”

We now have; Â

3yÂ +Â 5Â Â =Â 2(yÂ +Â 10)

**open the brackets**

â‡’ 3yÂ +Â 5 Â =Â 2yÂ +Â 20

**collect like terms**

â‡’ 3yÂ –Â 2y Â =Â 20Â – 5

y = 15

Substitute y Â =Â 15 into equation **(i)**Â Â “xÂ = Â 3**y**“

x = 3 (15)

x = 45

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

**Alternate Method**

Let the age of the father be x.

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

- The age of the father is thrice the sum of the age of his two sons:

_________Â xÂ = Â 3yÂ _________Â **(i)**

- 5 years hence: Father _____________ x + 5

The sons ____________ y + 5 + 5 + y + 10

- His age will be twice the sum of the ages of his two sons.

2(y + 10)

âˆ´Â Â Â xÂ +Â 5 Â = Â 2(yÂ +Â 10)Â __________________ Â **(ii)**

Substitute xÂ =Â 3yÂ in equation **(ii)**

To get 3y + 5 = 2y + 20

**Collect like terms**

3y – 2y = 20 – 5

y = 15

Substitute y Â =Â 15Â in equation **(i)**

x = 3y

x = 3 x 15

x = 45

Since x represents the age of the father,

The present age of the father is 45 years.

### Example 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

âˆ´ 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?Â ____Â Â Solve, 5(xÂ +Â 8)Â +Â 10 Â =Â 35

5(xÂ +Â 8)Â +Â 10 Â =Â 35

**Open the brackets**

5x + 40 + 10 = 35

5x + 50 = 35

**Collect like terms **

5x = 35 – 50

5x = -15

**Divide both sides by 5**

â‡’ \( \frac{5x}{5} = \frac{-15}{5} \\ \frac{\not{5}x}{\not{5}} = \frac{-15}{5} \\ \scriptsize x = \; -\; 3\)

The number is -3

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