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
2x22/7 = πx
x = (2x22/7) /π =(2x22/7) /(22/7)
x = 2
- find the mean proportion to :
(i) 8, 16
(ii) 0.3, 2.7
(ii)162/3 , 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
x2 = 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
x2 = 0.3 x 2.7 = 0.81
x = √(0.81) = 0.9
(ii)162/3 , 6
Let x be the mean proportion to 162/3 and 6
Then (162/3) /x = x/6
x2 = 162/3 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
x2 = 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) 31/3, 12/3, 21/2
(iii)15/7, 23/4, 33/5
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) 31/3, 12/3, 21/2
Let x be the fourth proportion
Then, 31/3: 12/3 :: 21/2: x
10/3 :5/3 : : 5/2 : x
10/3x = 5/3 x 5/2
10/3x = 25/6
x = 25/6 x 3/10 = 75/60 = 15/12 = 5/4
(iii)15/7, 23/14, 33/5
Let x be the fourth proportion
Then, 15/7: 23/14:: 33/5: 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 = 413/20
- Find the third proportion to:
(i) 12, 16
(ii) 4.5, 6
(iii) 51/2 , 161/2
Solution:
(i) 12, 16
Let x be the third proportion
Then 16:12 :: x : 16
12x = 16 x16
x = 16×16/12 = 64/3 = 211/3
(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) 51/2 , 161/2
Let x be the third proportion
Then 161/2: 51/2:: x : 161/2
11/2 x = 33/2 x 33/2
x = 33/2 x 33/2 x 2/11= 33×3/2 = 99/2 = 491/2
- In a map 1/4 cm represents 25km, if two cities are 21/2c apart on the map, what is the actual distance between them?
Solution:
Let 21/2 cm represemts x km
1/4cm: 25km :: 21/2cm : x km
1/4 x x = 25 x 21/2
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 = 93/8 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 t1/2 and C is t1/3 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. t1 = 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
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