Ratio and Proportion

Map Ratio

A model of a house is built in the ratio 1:50

(a) Find the length of a room in the model in cm, if actual length of the room is 3 m.

(b) Find the actual area of the room if it is 7 cm x 5 cm in the model.

Example

A map is drawn to a scale of 1 : 10000

i) Actual length of a road is 10 km, find its length on the map.

ii) Length of the road on the map is 20 cm. Find its actual length.

i) 10 km = 10,000 m = 1,000,000 cm

10000 units = 1,000,000

1 unit = 1,000,000/10,000 = 100 cm

ii) 1 unit = 20 cm

10000 units = 20 x 10,000 cm = 200,000 cm = 2000 m = 2 km

Direct Proportion

If a is proportional to b, we write a = kb

If a increases, b will increase.

If a decreases, b will decrease.

Example

a is proportional to b. If b = 10 when a = 25, write an equation connecting a and b.

Let a = kb

b = 10 when a = 25

25 = k x 10

k = 25/10

k = 2.5

Therefore a = 2.5 b

Example

a is proportional to b. The difference between values of a when b = 2 and b = 5 is 36.

a) Write an equation connecting a and b.

b) Find the value of a when b = 6.

c) Find the value of b when a = 24.

a) Let a = kb

When b = 2, a = k(2) = 2k

When b = 5, a = k(5) = 5k

5k – 2k = 36

3k = 36

k = 12

Therefore a = 12b

b) a = 12b

When b = 6, a = 12 x 6 = 72

c) a = 12b

When a = 24,

24 = 12 x b

b = 24/12 = 2

Inverse Proportion

If a inversely proportional to b, we write a = k/b

If a increases, b will decrease.

If a decreases, b will increase.

Example

a is inversely proportional to b. If b = 10 when a = 25, write an equation connecting a and b.

a = k/b

b = 10 when a = 25

25 = k/10

k = 25 x 10

k = 250

Therefore a = 250/ b

Example

a is inversely proportional to b, b = 8 when a = 4,

a) Write an equation connecting a and b.

b) Find the value of a when b = 4.

c) Find the value of b when a = 16.

a) Let a = k/b

b = 8 when a = 4

4 = k/8

k = 4 x 8 = 32

Therefore a = 32/b

b) a = 32/b

When b = 4, a = 32/4 = 8

c) a = 32/b

When a = 16

16 = 32/b

16 b = 32

b = 32/16 = 2

Questions

1. A Map is drawn to a scale of 1:50000. Find the actual length of a road in km that is represented by 7 cm on the Map.

2. Length of Running Track in a Map is 20 cm and its actual length is 200 m. Find the actual area of a Pond in m² which is 50 cm² in the Map.

3. a is directly proportional to b. a = 25 when b = 5. Find the value of a when b = 3

4. m is inversely proportional to n. m = 10 when n = 40. Find the value of m when n = 32

5. p is directly proportional to q². p = 12 when q = 2. Find the value of p when q = 3

6. r is inversely proportional to s³. r = 2 when s = 2. Find the value of r when s = 4

7. 10 painters are hired to a paint a building in 8 days. Find how many more painters required to paint the same building in 5 days.