Mathematics Form 1-4 : CHAPTER FIFTY TWO – COMPOUND PROPRTION AND RATES OF WORK

Specific Objectives

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

• Solve problems involving compound proportions using unitary and ratio methods;
• Apply ratios and proportions to real-life situations;
• Solve problems involving rates of work.

Content

• Proportional parts
• Compound proportions
• Ratios and rates of work
• Proportions applied to mixtures

Introduction

Compound Proportions

A proportion involving two or more quantities is called a compound proportion. Any four quantities a, b, c, and d are in proportion if;

Example

Find the value of a that makes 2, 5, a, and 25 be in proportion.

Solution

Since 2, 5, a, and 25 are in proportion, the cross products are equal:

Continued Proportions

In continued proportion, all the ratios between consecutive quantities are the same. For example:

P : Q, Q : R, R : S

10 : 5, 16 : 8, 4 : 2

Note that in the example, the ratio between different quantities, i.e., P:Q, Q:R, and R:S, are the same, i.e., 2:1 when simplified.

Continued proportion is very important when determining the net worth of individuals who own the same business or calculating the amounts of profit that different individual owners of a company or business should receive.

Proportional Parts

In general, if a quantity n is to be divided in the ratio a : b : c, then the parts of n proportional to a, b, and c are calculated accordingly.

Example

Omondi, Joel, and Cheroot shared sh 27,000 in the ratio 2:3:4 respectively. How much did each get?

Solution

The parts of sh 27,000 proportional to 2, 3, and 4 are calculated by dividing the total amount according to the ratio.

Example

Three people – John, Debby, and Dave contributed Ksh 119,000 to start a company. If the ratio of John’s contribution to Debby’s was 12:6 and the ratio of Debby’s to Dave’s was 8:4, determine the amount each partner contributed.

Solution

Ratio of John to Debby’s contribution = 12:6 = 2:1

Ratio of Debby to Dave’s contribution = 8:4 = 2:1

Since the ratios are in continued proportion, the combined ratio of John : Debby : Dave is calculated as follows:

John : Debby : Dave

12 : 6

8 : 4

Multiply the upper ratio by 8 and the lower ratio by 6:

John : Debby : Dave

96 : 48 : 24 = 4 : 2 : 1

The total ratio is 7.

The contributions are:

• John: (4/7) × 119,000 = Ksh 68,000
• Debby: (2/7) × 119,000 = Ksh 34,000
• Dave: (1/7) × 119,000 = Ksh 17,000

Example 2

You are presented with three numbers which are in continued proportion. If the sum of the three numbers is 38 and the product of the first and third numbers is 144, find the three numbers.

Solution

Let the three numbers in continued proportion (geometric progression) be a, ar, and ar², where a is the first number and r is the common ratio.

The sum is:

a + ar + ar² = 38 ………………………….. (1)

The product of the first and third numbers is:

a × ar² = 144

Or

a²r² = 144 ………………………………..(2)

Taking the square root of (2) gives:

ar = 12

Since the second number is 12, the sum of the first and third numbers is 26.

We now find two numbers whose sum is 26 and product is 144.

These numbers are 8 and 18.

Thus, the three numbers are 8, 12, and 18.

Verification:

8 + 12 + 18 = 38

8 × 18 = 144

The numbers are indeed in continued proportion.

Example

Given that x : y = 2 : 3, find the ratio (5x – 4y) : (x + y).

Solution

Since x : y = 2 : 3, let x = 2k and y = 3k.

Then, (5x – 4y) : (x + y) = (10k – 12k) : (2k + 3k) = -2k : 5k = -2 : 5.

Example

If show that.

Solution

Substituting kc for a and kd for b in the expression.

Therefore, the expression…

Rates of Work and Mixtures

Example

195 men working 10 hours a day can finish a job in 20 days. How many men are required to finish the job in 15 days if they work 13 hours a day?

Solution

Let x be the number of men required.

Using the formula for work done:

20 × 10 × 195 = 15 × 13 × x

Example

Tap P can fill a tank in 2 hours, and tap Q can fill the same tank in 4 hours. Tap R can empty the tank in 3 hours.

1. If tap R is closed, how long would it take taps P and Q to fill the tank?
2. Calculate how long it would take to fill the tank when taps P, Q, and R are all running.

Solution

1. Tap P fills 1/2 of the tank in 1 hour.
2. Tap Q fills 1/4 of the tank in 1 hour.
3. Tap R empties 1/3 of the tank in 1 hour.

In one hour, P and Q fill 1/2 + 1/4 = 3/4 of the tank.

Therefore, time taken to fill the tank = 1 ÷ (3/4) = 4/3 hours = 1 hour 20 minutes.

2. In 1 hour, P and Q fill 3/4 of the tank while R empties 1/3 of the tank.

When all taps are open, (3/4 – 1/3) = 5/12 of the tank is filled in 1 hour.

Time taken to fill the tank = 1 ÷ (5/12) = 12/5 = 2.4 hours = 2 hours 24 minutes.

Example

In what proportion should grades of sugar costing sh.45 and sh.50 per kilogram be mixed to produce a blend worth sh.48 per kilogram?

Solution

Method 1

Let n kilograms of the grade costing sh.45 per kg be mixed with 1 kilogram of the grade costing sh.50 per kg.

Total cost of the two blends is 45n + 50.

The mass of the mixture is n + 1.

Therefore, total cost of the mixture is 48(n + 1).

Equating the costs:

45n + 50 = 48n + 48

50 = 3n + 48

2 = 3n

n = 2/3

The two grades are mixed in the proportion 2 : 3.

Method 2

Let x kg of grade costing sh.45 per kg be mixed with y kg of grade costing sh.50 per kg. The total cost will be sh.(45x + 50y).

Cost per kg of the mixture is sh.48.

The proportion is x : y = 2 : 3.

End of topic

Past KCSE Questions on the Topic

1. Akinyi bought maize and beans from a wholesaler. She mixed the maize and beans in the ratio 4:3. She bought the maize at Kshs. 12 per kg and the beans at Kshs. 4 per kg. If she was to make a profit of 30%, what should be the selling price of 1 kg of the mixture?
2. A rectangular tank with base 2.4 m by 2.8 m and height 3 m contains 3,600 liters of water initially. Water flows into the tank at the rate of 0.5 litres per second. Calculate the time in hours and minutes required to fill the tank.
3. A company is to construct a parking bay with an area of 135 m². It is to be covered with a concrete slab of uniform thickness 0.15 m. To make the slab, cement, ballast, and sand are to be mixed so that their masses are in the ratio 1:4:4. The mass of 1 m³ of dry slab is 2,500 kg.

Calculate:

1. (i) The volume of the slab
2. (ii) The mass of the dry slab
3. (iii) The mass of cement to be used
4. If one bag of cement is 50 kg, find the number of bags to be purchased
5. If a lorry carries 7 tonnes of sand, calculate the number of lorries of sand to be purchased.

4. The mass of a mixture A of beans and maize is 72 kg. The ratio of beans to maize is 3:5 respectively.

(a) Find the mass of maize in the mixture.

(b) A second mixture B of beans and maize of mass 98 kg is mixed with A. The final ratio of beans to maize is 8:9 respectively. Find the ratio of beans to maize in B.

5. A retailer bought 49 kg of grade 1 rice at Kshs. 65 per kilogram and 60 kg of grade II rice at Kshs. 27.50 per kilogram. He mixed the two types of rice.

1. Find the buying price of one kilogram of the mixture.
2. He packed the mixture into 2 kg packets.

1. If he intends to make a 20% profit, find the selling price per packet.
2. He sold 8 packets and then reduced the price by 10% to attract customers. Find the new selling price per packet.
3. After selling 1/3 of the remainder at the reduced price, he raised the price to realize the original goal of 20% profit overall. Find the selling price per packet of the remaining rice.

6. A trader sells a bag of beans for Kshs. 1,200. He mixed beans and maize in the ratio 3:2. Find how much the trader should sell a bag of the mixture to realize the same profit.
7. Pipe A can fill an empty water tank in 3 hours while pipe B can fill the same tank in 6 hours. When the tank is full, it can be emptied by pipe C in 8 hours. Pipes A and B are opened at the same time when the tank is empty.

If one hour later, pipe C is also opened, find the total time taken to fill the tank.

8. A solution with volume 80 litres is made of 40% water and 60% alcohol. When x litres of water are 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.

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

9. A tank has two inlet taps P and Q and an outlet tap R. When empty, the tank can be filled by tap P alone in 4½ hours or by tap Q alone in 3 hours. When full, the tank can be emptied in 2 hours by tap R.

(a) The tank is initially empty. Find how long it would take to fill the tank:

1. If tap R is closed and taps P and Q are opened at the same time.
2. If all three taps are opened at the same time.

(b) The tank is initially empty and the three taps are opened as follows:

P at 8:00 a.m.

Q at 8:45 a.m.

R at 9:00 a.m.

(i) Find the fraction of the tank that would be filled by 9:00 a.m.

(ii) Find the time the tank would be fully filled.

10. Kipketer can cultivate a piece of land in 7 hours while Wanjiru can do the same work in 5 hours. Find the time they would take to cultivate the piece of land when working together.
11. Mogaka and Ondiso working together can do a piece of work in 6 days. Mogaka, working alone, takes 5 days longer than Ondiso. How many days does it take Ondiso to do the work alone?
12. 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) (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 A and 30 kg of milk from B.

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

13. A construction firm has two tractors T₁ and T₂. Both tractors working together can complete the work in 6 days while T₁ alone can complete the work in 15 days. After the two tractors had worked together for four days, tractor T₁ broke down.

Find the time taken by tractor T₂ to complete the remaining work.