## Rates

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

**Example**

If a car takes two hours to travel a distance of 160 km. then we will say that it is travelling at an average rate of 80 km per hour.

If two kilograms of maize meal is 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 1 20 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 = a^{mount 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 1 0 years old and his brother basher is 14 years old. Then alias age is
^{10}/_{1}_{4}of Bashir’s age, and their ages are said to be in the ratio of 10 to 14. Written, 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 2cm rep 5 km on the actual ground, then the ratio of map distance to map distance is 2 cm:5 x 100 000 cm, which is 2:500000.
- 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. If, example, 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 should be interpreted to mean^{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)

Consider the right hand side;

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, we 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 }x 20

=5x5

=25

**Example**

Decrease 45 in the ratio 7:9

**Solution**

New value =^{7}/_{9} x45

=7x5

=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 }x 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

- In order to compare ratios, they have to be expressed as fractions first, ie., a:b = a/b . the resultant fraction 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} = ^{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:

- A will be

a

a+b+c - B will be

b

a+b+c - C will be

c

a+b+c

**Example**

A 72-hactare 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 is 2+3+4= 9

The she shares are: ^{2}/_{9} x 72 ha = 16ha^{3}/_{9} x 72 ha= 24 ha^{4}/_{9} x72 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.

No. of cups 1 2 3 4 5 Cost (sh.) 20 40 60 80 100 - The ratio of the numbers 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 ratio and the number of cups, you should notice that they are always the same.
- If two quantities are such that when the one increases (decreases) in particular ratio, the other one also increases (decreases) in the ratio, the two quantities are said to be in direct proportion

**Example**

A car travels 40km 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 x ^{12}/_{5} km

**Example**

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

**Solution**

Time is decreased in the ratio 2:3 Speed must be increased in the ratio 3:2 average speed is 40 x ^{3}/_{2} km = 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 increased in the ratio 12:6.

Therefore, number of days decreases in the ratio 6:1 2.

Number of days taken =12 x ^{10}/_{8 }x ^{6}/_{12}

=7 ½ 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%:multiply by 100 .For example

0.125 = 0.125 ×1 00 = 12.5%^{2}/_{5}=^{2}/_{5}× 100 (i.e. of 100%) = 40%

Or^{2}/_{5}= 2 ÷ 5 × 100 = 40%

**Example**

Change ^{2}/_{5} into percentage.

**Solution**

= ^{2}/_{5} = ^{x}/_{100}

x = ^{2}/_{5} × 100

= 40%

**Example**

Convert 0.67 into a percentage: solution

0.67 = ^{67}/_{100}

As a percentage, 0.67= ^{67}/_{100 }x 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} x 100

= 80%

**Example**

A man earning sh. 4800 per month was given a 25% pay rise. What was his new salary?

**Solution**

New salary = ^{25}/_{100} x 4800 + 4 800

= 1200 + 4800

= sh. 6000

**Example**

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

**Solution**

Decrease in cost is 1200- 960= sh. 240

Percentage decrease = ^{240}/_{1200} x 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. 5600. 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 is ^{100}/_{112} x 5600 = sh. 5000^{John’s earnings}/_{Musa’s earnings} = ^{5}/_{3}

Musa’s earnings before the increase = ^{3}/_{5} x 5000 = sh. 3000

Musa’s new earnings = 9600 − 5 600 = sh. 4000

Musa’s change in earnings = 4000 − 1000 = sh. 3000

Percentage change in muse’s earnings = ^{1000}/_{3000} x1 00

=33^{1}/_{3} %

## Past KCSE Questions on the Topic

- Akinyi bought 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 42 per kg. If she was to make a profit of 30%. What should be the selling price of 1 kg of the mixture?
- 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?
- 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 1 50 respectively. A further Kshs 4000 per day was set aside for maintenance, insurance and loan repayment.
- How much money was collected from the passengers that day?
- How much was the net profit?

- How much money was collected from the passengers that day?
- On another day, the minibus was 80% full on the average for the three round trips, how much did each businessman get if the day’s profit was shared in the ratio 2:3?

- 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.
- Determine
- The total mass of milk fat in 50 kg of milk from farm A and 30 kg of milk from farm B
- The percentage of fat in a mixture of 50kg of milk A and 30kg of milk from B

- The total mass of milk fat in 50 kg of milk from farm A and 30 kg of milk from farm B
- The range of values of mass of milk from farm B that must be used in a 50kg mixture so that the mixture may have at least 4 percent fat.

- Determine
- In the year 2001, the price of a sofa set in a shop was Kshs 12, 000
- Calculate the amount of money received from the sale of 240 sofas sets that year
- In the year 2002 the price of each sofa set increased by 25% while the number of sets sold decreased by 1 0%. Calculate the percentage increase in the amount received from the sales
- If 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.

- 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 if the two years were equal.

- A solution whose volume is 80 litres is made up of 40% of water and 60% of alcohol. When x litres of water is added, the percentage of alcohol drops to 40%.
- Find the value of x
- Thirty litres of water is added to the new solution. Calculate the percentage of alcohol in the resulting solution
- If 5 litres of the solution in (b) above 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 resulting solution

- 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 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.
- Calculate the amount of money Cherop received more than Asha at the end of the first year.
- 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.

- Kipketer can cultivate a piece of land in 7 hrs 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.
- 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.
- A certain amount of money was shared among 3 children in the ratio 7:5:3 the largest share was Kshs 91 . Find the
- Total amount of money
- Difference in the money received as the largest share and the smallest share.

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