Example

**i. **If three-quarters of a certain number is 10 more than one-third of the number what is the number.

**Solution**

Let the number be x

\( \frac{3}{4}\scriptsize x = 10 \: +\: \normalsize \frac{1}{3}\scriptsize x \)**Take L.C.M**

**Cross multiply**

**Take like terms**

**Divide both sides by 5**

**ii.** A man spends one-third of his monthly salary earning on projects and one-quarter of the remainder on household needs. If he has N15000 left how much does he receive in a month?

**Solution**

**Let the income be x**

The remainder = \( \scriptsize \: 1 \: – \: \normalsize \frac{1}{3} \)

\( \scriptsize \: The\: remainder \: = \normalsize \frac{2}{3} \) \( \frac{1}{4} \scriptsize \: of \: the\; remainder \: on \: household \: needs\)= \( \frac{1}{4} \scriptsize \: of \: \normalsize \frac{2}{3} \)

= \( \frac{2}{12} \)

= \( \frac{1}{6} \)

In total He spent = Amount spent on projects + Amount spent on household needs

In total He spent = \(\frac{1}{3} \: + \: \frac{1}{6} \)

**Find the L.C.M**

= \(\frac{3}{6} \)

= \(\frac{1}{2} \)

Therefore he has \(\scriptsize 1 \: – \: \frac{1}{2} \) left

Which is equals to \(\frac{1}{2} \)

This means he has half his income, x, left which is also N15,000

\( \frac{1}{2}x = \scriptsize 15000 \) \( \scriptsize x = 2\: \times \: 15000 \) \( \scriptsize x = 30000 \)**iii.** Think of a number, treble it, then divide by 4, the result is 12 more than the original number, find the number.

**Solution**

Let the number be x

treble it = 3x

then divide by 4 = \( \frac{3x}{4} \)

the result is 12 more than the number = 12 + x

**Thereore,**

**Cross multiply**

**Take like terms**

**Divide both sides by -1**

**iv.** When 12 is subtracted from a number, and the answer is divided into 10, the result obtained is one-quarter of the original number. Find the number.

**Solution **

Let the number be x

\(\frac{x \: -\: 12}{10} = \frac{1}{4}\scriptsize x \)**Cross multiply**

**Bring like terms together**

**Divide both sides by -6**

**v.** A mother is four times as old as her daughter. If their age difference is three decades, find the ages of the mother and her daughter.

**Solution**

A decade is a period of 10 years.

Let *x* be the age of the daughter in years

Age** **of** **mother is therefore 4*x* (four times as old as her daughter)

Their age difference in years is 4*x* â€“* x*, i.e 3*x*

3 decades = 3 x 10 = 30

Therefore, 3*x* = 30

**Divide both sides by 3**

x = 10 years

**Age of mother** = 4x

= 4 x 10

= 40 years

**Age of daughter** = 10 years

**Work to do:**

**1.** When 9 is added to four times a certain number, the result is 37. Find the number.

**2. **When 3 is subtracted from eight times a number the result is 53.Â What is the number?

**3.** Three-quarter of a certain number is 18. Find the number.

**4.** The sum of two numbers is 13. If the two numbers differ by 3, find the smaller of the two numbers.Â Â

**5.** When a certain number is added to 2/9 of itself the result is 94 greater than the original number. Find the original number.

**Answers:**

**1.** 7**2.** 7**3. **24**4.** 5**5.** 32

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