After studying the chapter Ratio and Proportion and their general form; to understand and differentiate between different types of proportion; to acquire skills of writing proportion; to solve problems on time and work involving proportions; to apply proportion in day to day life situations.

## 2.4.1 Introduction to Ratio and Proportion

In a ratio a : b, the first term a is called the antecedent and the second term b is called the consequent. Ratio is an abstract quantity and has no unit. Ratio tells how many times the first term is there in the second term.

**Example 1: In the adjacent figure, find the ratio of the shortest side of the triangle to the longest side. **

Solution:

We see that the shortest side is of length 5 cm and the longest side is of length 13 cm. Hence the ratio is 5:13

**Example : Suppose the ratio of boys to girls in a school of 720 students is 7 : 5. How many more girls should be admitted to make the ratio 1 : 1?**

Solution:

For every 7 boys there are 5 girls. Thus out of 12 students, 7 are boys and 5 are girls. Hence the number of boys is ^{7}/_{12} x 720 = 420.

The number of girls is 720 – 420 = 300. Now we want the ratio of boys to girls to be 1:1. This means the number of boys and girls must be same. Since the deficiency of girls is 420 – 300 = 120, the school must admit 120 girls to make the ratio 1:1.

**Example 5: Consider the ratio 12:5 If this ratio has to be reduced by 20% which common number should be added to both the numerator and denominator?**

Solution:

Consider ^{12}/_{5}. This has to be reduced by 20%. This means we have to consider 80% of this number. Thus, we must get,

^{12}/_{5} x ^{80}/_{100} = ^{48}/_{25}

We have to find a such that,

^{12+a}/_{5+a} = ^{48}/_{25}

Cross multiplying

25(12 + a) = 48(5 + a)

48a – 25a = (25 x 12) – (48 x 5) =

23a = 60

a = ^{60}/_{23}

If we add ^{60}/_{23} to both terms of 12 : 5 we get a ratio which is 20% less than the original ratio.

### Ratio and Proportion – Exercise 2.4.1

**1.Write each of these ratios in the simplest form. **

**(i) 2:6**

**(ii)24:4**

**(iii) 14:21**

**(iv) 20: 100**

**(v) 18:24**

**(vi) 22:77**

**Solution: **

(i) 2:6 = 1:3 (dividing both by 2)

(ii) 24:4 = 6:1 (dividing both by 2)

(iii) 14:21 = 2:3 (dividing both by 7)

(iv) 20:100 = 1:5 (dividing both by 2)

(v) 18:24 = 3:4 (dividing both by 6)

(vi) 22:77 = 2:7 (dividing both by 11)

**2. A shop-keeper mixes 600 ml of orange juice with 900 ml of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in the fruit drink in its simplest form.**

**Solution: **

Ratio of volumes of

Orange juice and apple juice O:A

= 600:900

= 6:9

= 2:3

**3. a builder mixes 10 shovels of cement with 25 shovels of sand. Write the ratio of cement to sand. **

**Solution: **

Ratio of cement to sand = 10 shovels :25 shovels

**4.In a school there are 850 pupils and 40 teachers. Write the ratio of teachers to pupils. **

**Solution: **

Number of teachers : Number of pupils

= 40 : 850 = 4:85

**5. On a map, a distance of 5cm represent an actual distance of 15km. Write the ratio of the scale of the map. **

**Solution: **

Let x be the number to be added them

(49 + x) = (68 + x) = 3:4

4(49+X) = (68 + X)3

196+4X = 204 + 3X

4X – 3X = 204 – 196

x = 8

## 2.4.2 Proportion – Ratio and Proportion

### Ratio and Proportion – Exercise 2.4.2

**1. In the adjacent figure, two triangles are similar find the length of the missing side**

Solution:

Let the triangles be ABC and PQR

^{BC}/_{QR} = ^{AC}/_{PR}

^{5}/_{X} = ^{13}/_{39}

_{13X = 5 X 39}

_{X = }^{5X39}/_{13}^{ }= 5 X3 = 15

**What number is to 12 is 5 is 30?**

Solution

Let x be the number

x:12 :: 5 : 30

30x = 12×5

x = ^{12×5}/_{30 } = 2

**Solve the following properties:**

**(i). x : 5 = 3 : 6**

**(ii) 4 : y = 16 : 20**

**(iii) 2 : 3 = y : 9**

**(iv) 13 : 2 = 6.5 : x**

**(v) 2 : π = x : ^{22}/_{7}**

Solution:

(i). x : 5 = 3 : 6

6x = 5 x 3

6x = 15

x = ^{15}/_{6}

(ii) 4 : y = 16 : 20

4×20 = 16y

y = ^{4×20}/_{16}

y = 5

(iii) 2 : 3 = y : 9

2×9 = 3y

y = ^{2×9}/_{3} = 2×3 = 6

(iv) 13 : 2 = 6.5 : x

13x = 2 x 6.5

13x = 13

x = ^{13}/_{13} = 1

(v) 2 : π = x : ^{22}/_{7}

2x^{22}/_{7} = πx

x = (2x^{22}/_{7}) /_{π} =(2x^{22}/_{7}) /(^{22}/_{7})

x = 2

**find the mean proportion to :**

**(i) 8, 16**

**(ii) 0.3, 2.7**

**(ii)16 ^{2}/_{3} , 6**

**(iv) 1.25, 0.45**

Solution:

(i) 8, 16

Let x be the mean proportion to 8 and 16

Then ^{8}/_{x} = ^{x}/_{16}

x^{2} = 8 x 16 = 128

x = √128 = √(64×2) = 8√2

(ii) 0.3, 2.7

Let x be the mean proportion to 0.3 and 2.7

Then ^{0.3}/_{x} = ^{x}/_{2.7}

x^{2} = 0.3 x 2.7 = 0.81

x = √(0.81) = 0.9

(ii)16^{2}/_{3} , 6

Let x be the mean proportion to 16^{2}/_{3} and 6

Then (16^{2}/_{3}) /_{x} = ^{x}/_{6}

x^{2} = 16^{2}/_{3} x 6 = ^{50}/_{3} x 6 = 100

x = √100 = 10

(iv) 1.25, 0.45

Let x be the mean proportion to 1.25 and 0.45

Then ^{1.25}/_{x} = ^{x}/_{0.45}

x^{2} = 1.25 x 0.45

x = √(1.25 x 0.45) = √(1.25 x 0.45)x^{√(100×100)}/ _{√(100×100)}

= ^{√(125×45)}/_{√(100×100)} = ^{√(25x5x5x9)}/_{√(100×100)} = ^{5x5x3}/_{10×10}^{ }= ^{3}/_{4}

**Find the fourth proportion for the following:**

**(i) 2.8, 14, 3.5**

**(ii) 3 ^{1}/_{3}, 1^{2}/_{3}, 2^{1}/_{2}**

**(iii)1 ^{5}/_{7}, 2^{3}/_{4}, 3^{3}/_{5}**

Solution:

(i) 2.8, 14, 3.5

Let x be the fourth proportion

Then, 2.8 : 14 :: 3.5 : x

2.8x = 14×3.5

x = ^{14×3.5}/_{2.8} = 17.5

(ii) 3^{1}/_{3}, 1^{2}/_{3}, 2^{1}/_{2}

Let x be the fourth proportion

Then, 3^{1}/_{3}: 1^{2}/_{3} :: 2^{1}/_{2}: x

^{10}/_{3} :^{5}/_{3} : : ^{5}/_{2} : x

^{10}/_{3}x = ^{5}/_{3} x ^{5}/_{2}

^{10}/_{3}x = ^{25}/_{6}

x = ^{25}/_{6} x ^{3}/_{10} = ^{75}/_{60} = ^{15}/_{12} = ^{5}/_{4}

(iii)1^{5}/_{7}, 2^{3}/_{14}, 3^{3}/_{5}

Let x be the fourth proportion

Then, 1^{5}/_{7}: 2^{3}/_{14}:: 3^{3}/_{5}: x

^{12}/_{7} :^{31}/_{14} : : ^{18}/_{5} : x

^{12}/_{7 }x = ^{31}/_{14} x ^{18}/_{5}

^{12}/_{7 }x = ^{31×18}/_{14×5}

x = ^{31×18}/_{14×5} x ^{7}/_{12} = ^{31×3}/_{5×4} = ^{93}/_{20} = 4^{13}/_{20}

**Find the third proportion to:**

**(i) 12, 16**

**(ii) 4.5, 6**

**(iii) 5 ^{1}/_{2} , 16^{1}/_{2}**

Solution:

(i) 12, 16

Let x be the third proportion

Then 16:12 :: x : 16

12x = 16 x16

x = ^{16×16}/_{12} = ^{64}/_{3} = 21^{1}/_{3}

(ii) 4.5, 6

Let x be the third proportion

Then 6:4.5 :: x : 6

4.5x = 6 x6

x = ^{6×6}/_{4.5} = ^{36}/_{4.5} = ^{360}/_{45} = 8

(iii) 5^{1}/_{2} , 16^{1}/_{2}

Let x be the third proportion

Then 16^{1}/_{2}: 5^{1}/_{2}:: x : 16^{1}/_{2}

^{11}/_{2} x = ^{33}/_{2} x ^{33}/_{2}

x = ^{33}/_{2} x ^{33}/_{2} x ^{2}/_{11}= ^{33×3}/_{2} = ^{99}/_{2} = 49^{1}/_{2}

- I
**n a map**^{1}/_{4 }cm represents 25km, if two cities are 2^{1}/_{2}c apart on the map, what is the actual distance between them?

Solution:

Let 2^{1}/_{2 }cm represemts x km

^{1}/_{4}cm: 25km :: 2^{1}/_{2}cm : x km

^{1}/_{4} x x = 25 x 2^{1}/_{2}

^{x}/_{4} = ^{25 x5 }/_{2}

x = ^{25×5}/_{2} x 4 = 25x5x2 = 250km

**Suppose 30 out of 500 components for a computer were found defective. At this rate how many defective components would he found in 1600 components?**

Solution:

Number of defective components in 500 components = 30

Let x be the number of defective components in 1600 components

then 30:500 :: x :1600

30×1600 = 500x

x = ^{30×1600}/_{500 }=96

## 2.4.3 Time and Work – Ratio and Proportion

### Ratio and Proportion – Exercise 2.4.3

**Suppose A and B together can do a job in 12 days, while B alone can finish a job in 24 days. In how many days can A alone finish the work?**

**Solution: **

Number of days in which A and B together can finish the work = 12 days

Number of days in which B alone can finish the work = 30

^{1}/_{T} = ^{1}/_{m} + ^{1}/_{n}

^{1}/_{12} = ^{1}/_{m} + ^{1}/_{30}

^{1}/_{m} = ^{1}/_{30} – ^{1}/_{12} = ^{5-2}/_{60} = ^{3}/_{60} = ^{1}/_{20}

A can finish the work in 20 days.

Suppose A is twice as good a workman as B and together they can finish a job in 24 days. How many days A alone takes to finish the job?

Solution:

A is twice as good a workman as B

i.e if B can finish a work in t days A can finish it in ^{1}/_{2} days

^{1}/_{T} = ^{1}/_{m} + ^{1}/_{n}

^{1}/_{24} = ^{1}/_{t/2} + ^{1}/_{t} = ^{2}/_{t} + ^{1}/_{t} = ^{3}/_{t}

^{1}/_{24} = ^{3}/_{t}

t = 24 x 3 = 72

i.e, B takes 72days to finish the job

A takes ^{72}/_{2} = 36 days to finish it

**Suppose B is 60% more efficient them A. if A can finish a job in 15 days how many days B needs to finish the same job?**

Solution:

A can finish a work in 15 days.

Work done A in 1 day = ^{1}/_{15}

B is 60% more efficient

Work done by B in 1 day

^{1}/_{15} + ^{1}/_{15} x ^{60}/_{100}

= ^{1}/_{15} (1 + ^{60}/_{100})

= ^{1}/_{15} ( ^{8}/_{5})

= ^{8}/_{75}

Number of days in which B alone can finish the work = ^{1}/_{(8/75)} = ^{75}/_{8} = 9^{3}/_{8} days

**Suppose A can do a piece of work in 14 days while B can do it in 21 days. They begin together and worked at it for 6 days. Then A fell ill B had to complete the work alone. In how many days was the work completed?**

Solution:

M = 14 days

N = 21 days

Part of work done in 6 days

= (^{1}/_{14} + ^{1}/_{21})^{6}

= 6(^{3+2}/_{42}) = ^{5×6}/_{42} = ^{5}/_{7}

Remaining part of the work = ^{1-5}/_{7} = ^{2}/_{7}

Days taken by B to finish

^{2}/_{7} part of the work = ^{(2/7)}/_{(1/21)} = ^{2}/_{7} x ^{21}/_{7} = 6 days

Total number of days in which the work is completed = 6+6 = 12 days

**Suppose A takes twice as much time as B and thrice as much time as C to complete a work. If all of them work together they can finish the work in 2 days. How much time B and C working together will take to finish it?**

Solution:

If A alone takes to t1 days to do the work , B finishes it in t_{1}/2 and C is t_{1}/_{3} days

^{1}/_{T} = ^{1}/_{t1} +^{1}/_{t2} + ^{1}/_{t3}

= ^{1}/_{t1} +^{1}/_{(t1/2)} + ^{1}/_{(t1/3)}

= ^{1}/_{t1} +^{2}/_{t1} + ^{3}/_{t1}

= ^{6}/_{t1}

^{1}/_{T} = ^{1}/_{2}

^{1}/_{2} = ^{6}/_{T1}

i.e. t_{1} = 12 dyas

B takes ^{12}/_{2} = 6days

C takes ^{12}/_{3} = 4 days

Part of work done by B

In one day = ^{1}/_{6}

Part of work done by C in one day = ^{1}/_{4}

If B and C together takes t days to finish the work ^{1}/_{T} = ^{1}/_{6} + ^{1}/_{4} = ^{2+3}/_{12} = ^{5}/_{12}

T = ^{12}/_{5} = 2.4 days