A family spends 3/7of their monthly income on food,5/14 on clothing and 1/5 on rent.
What fraction of the family's income is
spent on other things? If in a particular
month, the family spends GH¢2,400 on
other things, how much did the family
spend on clothing? What was the family's
income for the month.​

Suppose that the family's monthly income is X.

we know that they spend:

(3/7)*X on food.

(5/14)*X on clothing

(1/5)*X on rent.

a) what fraction of the family's income is spent in other things?

To solve this, we assume that N is spent on other things, then we should have that:

(3/7)*X + (5/14)*X + (1/5)*X + N*X = X

We can divide both sides by X to remove it, so we get:

(3/7) + (5/14) + (1/5) + N = 1

we can solve this for N.

3/7 + 5/14 + 1/5 + N = 1

6/14 + 5/14 + 1/5 + N = 1

(6 + 5)/14 + 1/5 + N = 1

11/14 + 1/5 + N = 1

The smallest common factor of 14 and 5 is 14*5 = 70

Then we need to multiply the first fraction by (5/5) and the second by (14/14)

(5/5)*(11/14) + (14/14)*(1/5) + N = 1

55/70 + 14/70 + N = 1

69/70 + N = 1

N = 1 - 69/70 = 1/70

N = 1/70

This means that they spent 1/70 of their monthly income on other things.

b) One particular month, the family spends  GH¢2,400 on other things, then if the monthly income is X, we have:

(1/70)*X = GH¢2,400

X = 70* GH¢2,400 = GH¢168,000

This means that the monthly income of the family is GH¢168,000

And we know that they spend (5/14)*X on clothing, then this month they spent:

(5/14)*GH¢168,000 = GH¢60,000 on clothing.

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