Proportions in mathematics has to do with the method of representing the comparisons of two or more quantities and it usually contains units.

## There are three types of proportions in mathematics, which are:

### Direct or Simple proportion:

This type of proportion compares two quantities in a way that when one quantity increases, the other quantity also increases and when it decreases, the other decreases as well.

### Inverse Proportion:

This type of proportion compares two quantities in such a way that when one quantity increases, the other decreases and vise versa.

### Complex Proportion:

This is the combination of either two or more Direct or Inverse Proportions, it may as well contain both direct and inverse Proportions with more than two quantities.

## QUESTIONS AND SOLUTIONS:

### Q1: 9 men do a piece of work in 28 days.

### (a) How long will it take 14 men to do that same work?

### (b) How many men will do it in 56 days?

**Solution**:

(1a) 9 men = 28 days

One man can do the work in (28 x 9) days = 252 days.

Therefore: 14 men can finish the work in (252/14) days = 18 days

(b) How many men will do it in 56 days?

Solution:

Let the total number of men that can do the work in 56 days be x.

9 men do it in 28 days

x men do it in 56 days

That is:

9 men = 28 days

x men = 56 days

Cross multiply

28x = 9 x 56

28x = 504

Divide both sides by the coefficient of x

x = 504/28

x = 18 men.

### Q2: P and Q are partners in a venture. P contributed #20,000 for nine months and Q contributed #50,000 for one year. Find each person’s share of profit of #6,300.

**Solution**:

P = #20,000 for nine months

Q = #50,000 for one year(12 months)

Ratio at which they contributed the money P:Q = 9:12 = 3:4

Total income = #6,300

Add the ratio together :-

3 + 4 = 7

P is collecting 3/7 x #6,300 = #2,700

Q is collecting 4/7 x #6,300 = #3,600

### Q3: A factory worker is paid #269.50 in 7 days working 11 hours a day. How much must be paid to him for 26 days working 9 hours a day?

**Solution**:

Let the amount he will be paid in 26 days working 9 hours be x.

Total duration of his work that fetched him #269.5 = number of days multiply by the hours he worked

= 7 x 11 = 77 hours

77 hours = #269.5

Total duration of his work that will fetch him #x = number of days multiply by the hours he worked

= 26 x 9 = 234 hours

234 hours = #x

So,

77 hours = #269.5

234 hours = #x

Cross multiply

77x = 269.5 x 234

77x = 63,063

x = #819.00

### Q4: 24 men can plough 42 hectares of land in 14 days. How many days will it take 16 men to plough 64 hectares?

**Solution**:

If 24 men can plough 42 hectares of land in 14 days,

one man can plough the same 42 hectares in (14 x 24)/42

= 336/42

= 8 days.

Therefore 16 men can plough 64 hectares in 8 x (64/16)

= 8 x 4 = 32 days.

### Q5: A motorist drives 80km per hour in 5 hours and another 60km per hour in 4 hours. Find the average speed for the whole journey.

**Solution**:

Speed = Distance/Time

For first journey:

80 = D/5

Distance = D = 80 x 5

= 400km

For second journey:

D = 60 x 4 = 240km

Therefore, average speed

= (D1 + D2)/(T1 + T2)

= (400 + 240)/(5 + 4)

= 640/9

= 71.11km per hour

### Q6: Nneka is half as old as Joke and Joke is half as old as Zainab. The sum of their ages is 336 years. Find their ages.

**Solution**:

Let Zainab’s age be “y”

Joke’s age = y/2

Nneka’s age = y/2 ÷ 2

= y/2 x 1/2 = y/4

The sum of their ages

= y + y/2 + y/4 = 336

Solve for “y”

7y = 1344

Divide both sides by the coefficient of y

y = 1344/7

y = 192 years = Zainab’s age

Joke’s age = 192/2 = 96 years

Nneka’s age = 96/2 = 48 years.

### Q7: If 10 students consume 1185kg of meat in 21 days, find how much 16 students will consume in 14 days if consumption takes the same rate.

**Solution**:

Let the amount that 16 students consume in 14 days be x.

10 students consume 1185kg of meat in 21 days

1 student consumes (10 x 21)/1185 days.

16 students will consume xkg in 14 days

So,

[(21 x 10)/1185] X (x/16) = 14

210x = 1185 x 16 x 14

210x = 265,440

x = 265,440/210

x = 1264kg

### Q8: The cost of feeding 24 goats for 12 days is #360. Find the number of days that 20 goats will be fed with #400.

**Solution**:

Let the number of days be y.

If the cost of feeding 24 goats for 12 days is #360,

The cost of feeding 1 goat for 1 day

= (12 x 24)/ 360 = 288/360.

And the cost of feeding 20 goats in y days = 400

Again, the cost of feeding 1 goat for one day = (20 x y)/400

= 20y/400 = y/20

Equate both of them together and solve for y.

y/20 = 288/360

360y = 5,760

y = 16 days

### Q9: Divide #738 in the ratio 3/2 : 2 : 10/3

Solution:

First add up the ratio

3/2 + 2 + 10/3 = 41/6

For portion 3/2

3/2 ÷ 41/6 x 738

= 3/2 x 6/41 x 738 = (3 x 3 x 738)/41

= 6642/41

= #162

For portion 2

2 ÷ 41/6 x 738

= 2 x 6/41 x 738 = (2 x 6 x 738)/41

= 8856/41

= #216

For portion 10/3

10/3 ÷ 41/6 x 738

= 10/3 x 6/41 x 738 = (10 x 2 x 738)/41

= 14760/41

= #360

### Q10: 5kg of ginger costing #24 per kg is diluted with 10kg of garlic costing #18 per kg. Find the cost of the mixture per kg.

Solution:

Cost of 5kg of ginger = 5 x #24 = #120

Cost of 10kg of garlic = 10 x #18 = #180

Total cost of the two = #120 + #180 = #300

Total kg of the two = 5kg + 10kg = 15kg

Therefore, the cost of the mixture per kg = #300/15kg

= #20 per kg