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Specific Objectives

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

(a) Solve problems involving compound proportions using unitary and ratiomethods;

(b) Apply ratios and proportions to real life situations;

(c) Solve problems involving rates of work.

Content

(a) Proportional parts

(b) Compound proportions

(c) Ratios and rates of work

(d) Proportions applied to mixtures.

Introduction

Compound proportions

The proportion involving two or more quantities is called 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 to be in proportion;

Solution

Since 2 , 5 ,a , and 25 are in proportion

Continued proportions

In continued proportion, all the ratios between different quantities are the same; but always remember that the relationship exists between two quantities 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 even calculating the amounts of profit that different individual owners of a company or business should take home.

Proportional parts

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

Example

Omondi, Joel, 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, 4 are

Example

Three people – John, Debby and Dave contributed ksh 119, 000 to start a company. If the ratio of the contribution of John to Debby was 12:6 and the contribution of Debby to Dave was 8:4, determine the amount in dollars that every 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

As you can see, the ratio of the contribution of John to Debby and that of Debby to Dave is in continued proportion.

Hence

To determine the ratio of the contribution between the three members, we do the calculation as follows:

John: Debby: Dave

12 : 6

8 : 4

We multiply the upper ratio by 8 and the lower ratio by 6, thus the resulting ratio will be:

John: Debby: Dave

96: 48 : 24

= 4 : 2 : 1

The total ratio = 7

The contribution of the different members can then be found as follows:

John

Debby

Dave

John contributed ksh 68, 000 to the company while Debby contributed ksh 34, 000 and Dave contributed 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 number and the third number is 144, find the three numbers.

Solution

Let us assume that the three numbers in continued proportion or Geometric Proportion are a, ar and a where a is the first number and r is the rate.

a+ar+a = 38 ………………………….. (1)

The product of the 1^{st} and 3^{rd} is

a × a = 144

Or

(ar)2 = 144………………………………..(2)

If we find the square root of (a, then we will have found the second number:

=

Since the value of the second number is 12, it then implies that the sum of the first and the third number is 26.

We now proceed and look for two numbers whose sum is 26 and product is 144.

Clearly, the numbers are 8 and 18.

Thus, the three numbers that we were looking for are 8, 12 and 18.

Let us work backwards and try to prove whether this is actually true:

8 + 12 + 18 = 18

What about the product of the first and the third number?

8 × 18 = 144

What about the continued proportion

The numbers are in continued proportion

Example

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

Solution

Since x: y =2: 3

(5x – 4y): (x + y) = (10k – 12 k) 🙁 2k + 3 k)

=-2k: 5k

= – 2: 5

Example

If show that.

Solution

Substituting kc for a and kd for b in the expression

Therefore expression

Rates of work and mixtures

Examples

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

Solution:

Let x be the no. of men required

Days hours Men

20 10 195

15 13 x

20 x 10 x 195

Example

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

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

Solution

- Tap P fills of the tank in 1 h.

Tap Q fills of the tank in 1 h.

Tap R empties of the tank in 1 h.

In one hour, P and Q fill

Therefore

Time taken to fill the tank

- In 1 h, P and Q fill of tank while R empties of the tank.
When all taps are open ,of the tank is filled in 1 hour.

= 2

Example

In what proportion should grades of sugars costing sh.45 and sh.50 per kilogram be mixed in order 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 grade costing sh.50 per kg.

Total cost of the two blends is sh.

The mass of the mixture is

Therefore total cost of the mixture is

45n + 50 = 48 (n +1)

45n + 50 = 48 n + 48

50 = 3n + 48

2 = 3n

The two grades are mixed in the proportion

**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 + 50 y)

Cost per kg of the mixture is sh.

The proportion is x : y = 2:3

End of topic

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Past KCSE Questions on the topic.

1. Akinyi bought and beans from a wholesaler. She then mixed the maize and beans the ratio 4:3 she brought the maize as Kshs. 12 per kg and the beans 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 of base 2.4 m by 2.8 m and a height of 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 whose area is 135m^{2}. It is to be covered with concrete slab of uniform thickness of 0.15. 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 m^{3} of dry slab is 2, 500kg.

Calculate

(a) (i) The volume of the slab

(ii) The mass of the dry slab

(iii) The mass of cement to be used

(b) If one bag of the cement is 50 kg, find the number of bags to be purchased

- 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 of B of beans and maize of mass 98 kg in 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 tow types of rice.

- Find the buying price of one kilogram of the mixture
- He packed the mixture into 2 kg packets
- If he intends to make a 20% profit find the selling price per packet
- He sold 8 packets and then reduced the price by 10% in order to attract customers. Find the new selling price per packet.
- After selling
^{1}/_{3}of the remainder at reduced price, he raised the price so as 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 ration 3: 2. Find how much the trader should he 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 whose volume is 80 litres is made 40% of water and 60% of alcohol. When 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

(c) If 5 litres of the solution in (b) is added to 2 litres of the original solution, calculate in the simplest form, the ratio of water to that of 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 4 ½ 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 up the tank

- If tap R is closed and taps P and Q are opened at the same time (2mks)
- If all the 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 up

10. Kipketer can cultivate a piece of land in 7 hrs 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 Onduso. How many days does it take Onduso to do the work alone.

12. Wainaina has two dairy farms A and B. Farm A produces milk with 3 ¼ percent fat and farm B produces milk with 4 ¼ percent fat.

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

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 may have at least 4 percent fat.

13. A construction firm has two tractors T_{1} and T_{2.} Both tractors working together can complete the work in 6 days while T_{1} 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 T_{2} complete the remaining work.