Mathematics Form 1-4 : CHAPTER ELEVEN – RATES, RATIO, PROPORTION AND PERCENTAGE

Specific Objectives

By the end of the topic the learner should be able to:

1. Define rates
2. Solve problems involving rates
3. Define ratio
4. Compare two or more quantities using ratios
5. Change quantities in a given ratio
6. Compare two or more ratios
7. Represent and interpret proportional parts
8. Recognize direct and inverse proportions
9. Solve problems involving direct and inverse proportions
10. Convert fractions and decimals to percentages and vice versa
11. Calculate percentage change in a given quantity
12. Apply rates, ratios, percentages to real life situations and proportion

Content

1. Rates
2. Solving problems involving rates
3. Ratio
4. Comparing quantities using ratio
5. Increase and decrease in a given ratio
6. Comparing ratios
7. Proportion: direct and inverse
8. Solve problems on direct and inverse proportions
9. Fractions and decimals as percentages
10. Percentage increase and decrease
11. Application of rates, ratios, percentages and proportion to real life situations

Introduction

Rates

A rate is a measure of quantity, comparing one quantity with another of a different kind.

Example

If a car takes two hours to travel a distance of 160 km, then we say that it is travelling at an average rate of 80 km per hour. If two kilograms of maize meal are sold for sh. 38.00, then we say that maize meal is selling at the rate of sh. 19.00 per kilogram.

Example

What is the rate of consumption per day if twelve bags of beans are consumed in 120 days?

Solution

Rate of consumption = number of bags / number of days = 12 / 120 = 1/10 bags per day

Example

A laborer’s wage is sh. 240 per eight hours working day. What is the rate of payment per hour?

Solution

Rate = amount of money paid / number of hours = 240 / 8 = sh. 30 per hour

Ratio

A ratio is a way of comparing two similar quantities. For example, if Alias is 10 years old and his brother Bashir is 14 years old, then Alias’s age is 10/14 of Bashir’s age, and their ages are said to be in the ratio of 10 to 14, written as 10:14.

Alias age : Bashir’s age = 10:14

Bashir’s age : Alias age = 14:10

In stating a ratio, the units must be the same. If on a map 2 cm represents 5 km on the actual ground, then the ratio of map distance to actual distance is 2 cm : 5 × 100,000 cm, which is 2 : 500,000.

A ratio is expressed in its simplest form in the same way as a fraction, e.g., 10/14 = 5/7, hence 10:14 = 5:7. Similarly, 2:500,000 = 1:250,000.

A proportion is a comparison of two or more ratios. For example, if a, b, and c are three numbers such that a : b : c = 2 : 3 : 5, then a, b, c are said to be proportional to 2, 3, 5 and the relationship means a/2 = b/3 = c/5.

Similarly, we can say that a : b = 2 : 3, b : c = 3 : 5, a : c = 2 : 5.

Example 3

If a : b = 3 : 4 and b : c = 5 : 7, find a : c.

Solution

a : b = 3 : 4 (i)

b : c = 5 : 7 (ii)

Multiply (i) by 5 and (ii) by 4 to get a : b = 15 : 20 and b : c = 20 : 28.

Thus, a : b : c = 15 : 20 : 28 and a : c = 15 : 28.

Increase and decrease in a given ratio

To increase or decrease a quantity in a given ratio, express the ratio as a fraction and multiply it by the quantity.

Example

Increase 20 in the ratio 4 : 5.

Solution

New value = (5/4) × 20 = 25

Example

Decrease 45 in the ratio 7 : 9.

Solution

New value = (7/9) × 45 = 35

Example

The price of a pen is adjusted in the ratio 6 : 5. If the original price was sh. 50, what is the new price?

Solution

New price : old price = 6 : 5

New price / old price = 6/5

New price = (6/5) × 50 = sh. 60

Note: When a ratio expresses a change in a quantity (an increase or decrease), it is usually put in the form of new value : old value.

Comparing ratios

To compare ratios, express them as fractions first, i.e., a : b = a/b. The resultant fractions can then be compared.

Example

Which ratio is greater, 2 : 3 or 4 : 5?

Solution

2 : 3 = 2/3, 4 : 5 = 4/5

2/3 = 10/15, 4/5 = 12/15, so 4/5 > 2/3

Thus, 4 : 5 > 2 : 3.

Distributing a quantity in a given ratio

If a quantity is to be divided in the ratio a : b : c, the fraction of the quantity represented by:

1. A will be a / (a + b + c)
2. B will be b / (a + b + c)
3. C will be c / (a + b + c)

Example

A 72-hectare farm is to be shared among three sons in the ratio 2 : 3 : 4. What will be the sizes in hectares of the three shares?

Solution

Total number of parts = 2 + 3 + 4 = 9

The shares are:

2/9 × 72 ha = 16 ha

3/9 × 72 ha = 24 ha

4/9 × 72 ha = 36 ha

Direct and inverse proportion

Direct proportion

The table below shows the cost of various numbers of cups at sh. 20 per cup.

The ratio of the number of cups in the fourth column to the number of cups in the second column is 4 : 2 = 2 : 1. The ratio of the corresponding costs is 80 : 40 = 2 : 1. By considering the ratio of costs in any two columns and the corresponding number of cups, you should notice that they are always the same.

If two quantities are such that when one increases (decreases) in a particular ratio, the other also increases (decreases) in the same ratio, they are said to be directly proportional.

Example

A car travels 40 km on 5 litres of petrol. How far does it travel on 12 litres of petrol?

Solution

Petrol is increased in the ratio 12 : 5.

Distance = 40 × (12/5) km = 96 km.

Example

A train takes 3 hours to travel between two stations at an average speed of 40 km per hour. At what average speed would it need to travel to cover the same distance in 2 hours?

Solution

Time is decreased in the ratio 2 : 3.

Speed must be increased in the ratio 3 : 2.

Average speed = 40 × (3/2) km/h = 60 km/h.

Example

Ten men working six hours a day take 12 days to complete a job. How long will it take eight men working 12 hours a day to complete the same job?

Solution

Number of men decreases in the ratio 8 : 10.

Therefore, the number of days taken increases in the ratio 10 : 8.

Number of hours increases in the ratio 12 : 6.

Therefore, the number of days decreases in the ratio 6 : 12.

Number of days taken = 12 × (10/8) × (6/12) = 7½ days.

Percentages

A percentage (%) is a fraction whose denominator is 100. For example, 27% means 27/100.

Converting fractions and decimals into percentages

To write a decimal or fraction as a percentage, multiply by 100. For example:

0.125 = 0.125 × 100 = 12.5%

2/5 = (2 ÷ 5) × 100 = 40%

Example

Change 2/5 into percentage.

Solution

(2 ÷ 5) × 100 = 40%

Example

Convert 0.67 into a percentage.

Solution

0.67 × 100 = 67%

Percentage increase and decrease

A quantity can be expressed as a percentage of another by first writing it as a fraction of the given quantity.

Example

A farmer harvested 250 bags of maize in a season. If he sold 200 bags, what percentage of his crops does this represent?

Solution

Let x be the percentage sold.

Then, x/100 = 200 / 250

So, x = (200 / 250) × 100 = 80%

Example

A man earning sh. 4,800 per month was given a 25% pay rise. What was his new salary?

Solution

New salary = (25/100 × 4800) + 4800 = 1200 + 4800 = sh. 6,000

Example

A dress which was costing sh. 1,200 now goes for sh. 960. What is the percentage decrease?

Solution

Decrease in cost = 1,200 – 960 = sh. 240

Percentage decrease = (240 / 1,200) × 100 = 20%

Example

The ratio of John’s earnings to Muse’s earnings is 5 : 3. If John’s earnings increase by 12%, his new figure becomes sh. 5,600. Find the corresponding percentage change in Muse’s earnings if the sum of their new earnings is sh. 9,600.

Solution

John’s earnings before the increase = (100 / 112) × 5600 = sh. 5,000

John’s earnings : Muse’s earnings = 5 : 3

Muse’s earnings before the increase = (3/5) × 5000 = sh. 3,000

Muse’s new earnings = 9,600 – 5,600 = sh. 4,000

Muse’s change in earnings = 4,000 – 3,000 = sh. 1,000

Percentage change in Muse’s earnings = (1,000 / 3,000) × 100 = 33%

End of topic

Past KCSE Questions on the topic

1. Akinyi bought maize and beans from a wholesaler. She then mixed the maize and beans in the ratio 4 : 3. She bought the maize at Kshs 21 per kg and the beans at 42 per kg. If she was to make a profit of 30%, what should be the selling price of 1 kg of the mixture?
2. Water flows from a tap at the rate of 27 cm³ per second into a rectangular container of length 60 cm, breadth 30 cm, and height 40 cm. If at 6:00 PM the container was half full, what will be the height of water at 6:04 PM?
3. Two businessmen jointly bought a minibus which could ferry 25 paying passengers when full. The fare between two towns A and B was Kshs 80 per passenger for one way. The minibus made three round trips between the two towns daily. The cost of fuel was Kshs 1,500 per day. The driver and the conductor were paid daily allowances of Kshs 200 and Kshs 150 respectively.

(a) (i) How much money was collected from the passengers that day?

(ii) How much was the net profit?

(b) On another day, the minibus was 80% full on average for the three round trips. How much did each businessman get if the day’s profit was shared in the ratio 2 : 3?

4. Wainaina has two dairy farms, A and B. Farm A produces milk with 3¼ percent fat and farm B produces milk with 4¼ percent fat.

(a) Determine

(i) The total mass of milk fat in 50 kg of milk from farm A and 30 kg of milk from farm B

(ii) The percentage of fat in a mixture of 50 kg of milk from farm A and 30 kg of milk from farm B

(b) The range of values of mass of milk from farm B that must be used in a 50 kg mixture so that the mixture may have at least 4 percent fat.

5. In the year 2001, the price of a sofa set in a shop was Kshs 12,000.

(a) Calculate the amount of money received from the sales of 240 sofa sets that year.

(b) (i) In the year 2002 the price of each sofa set increased by 25% while the number of sets sold decreased by 10%. Calculate the percentage increase in the amount received from the sales.

(ii) If at the end of year 2002, the price of each sofa set changed in the ratio 16 : 15, calculate the price of each sofa set in the year 2003.

(c) The number of sofa sets sold in the year 2003 was P% less than the number sold in the year 2001. Calculate the value of P, given that the amounts received from sales in the two years were equal.

6. A solution whose volume is 80 litres is made up of 40% water and 60% alcohol. When x litres of water is added, the percentage of alcohol drops to 40%.

(a) Find the value of x.

(b) Thirty litres of water is added to the new solution. Calculate the percentage of alcohol in the resulting solution.

(c) If 5 litres of the solution in (b) above is added to 2 litres of the original solution, calculate in simplest form the ratio of water to alcohol in the resulting solution.

7. Three business partners, Asha, Nangila, and Cherop contributed Kshs 60,000, Kshs 85,000, and Kshs 105,000 respectively. They agreed to put 25% of the profit back into the business each year. They also agreed to put aside 40% of the remaining profit to cater for taxes and insurance. The rest of the profit would then be shared among the partners in the ratio of their contributions. At the end of the first year, the business realized a gross profit of Kshs 225,000.

(a) Calculate the amount of money Cherop received more than Asha at the end of the first year.

(b) Nangila further invested Kshs 25,000 into the business at the beginning of the second year. Given that the gross profit at the end of the second year increased in the ratio 10 : 9, calculate Nangila’s share of the profit at the end of the second year.

8. Kipketer can cultivate a piece of land in 7 hours while Wanjiku can do the same work in 5 hours. Find the time they would take to cultivate the piece of land when working together.
9. Mogaka and Onduso working together can do a piece of work in 6 days. Mogaka working alone takes 5 days longer than Onduso. How many days does it take Onduso to do the work alone?
10. A certain amount of money was shared among 3 children in the ratio 7 : 5 : 3. The largest share was Kshs 91. Find the

(a) Total amount of money.

(b) Difference in the money received as the largest share and the smallest share.