Exercise 8 A
Question 1
Express each of the following ratios in the simplest form:
(i) $28: 63$
Sol: $\frac{28}{63}$
= 4: 9
(ii) 105: 133
Sol: $\frac{105}{133}$
= 15: 19
(iii) 0.4: 0.6 66
Sol: $\frac{0 \cdot 4}{0 \cdot 6}$
= 2: 366
(iv) $2.7: 9$
Sol: $\frac{27}{90}$
=3:10
(v) $\frac{1}{4}: \frac{1}{9}$
Sol: $\frac{1}{4} \div \frac{1}{9}$
= $\frac{1}{4} \times \frac{9}{1} \quad \Rightarrow 9: 4$
(vi) $3 \frac{1}{2}: 2 \frac{1}{4}$
Sol: $\frac{7}{2} \div \frac{9}{4}$
= $\frac{7}{2} \times \frac{4}{9}$
= 14: 9
(vii) $\frac{1}{4}: \frac{1}{8}: \frac{1}{10}$
Sol: L.C.M of 4, 8 , 10 is = 40
$\frac{1}{4} \times 40: \frac{1}{8} \times 40: \frac{1}{10} \times 40=10: 5: 4$
(viii) $1 \frac{3}{4}: 2 \frac{2}{3}: 1 \frac{5}{6}$
Sol: $\frac{7}{4}: \frac{8}{3}: \frac{11}{6}$
(xi) $125 \mathrm{~g}: 1 \mathrm{~kg}$
Sol: $\frac{125}{1000}$
=1: 8
(x) 4 days : 2 weeks
4 days: 14 days
Sol: $\frac{4}{14}$
= 2: 7
(xi) 36 minutes to $1 \frac{1}{2}$ Hours
Sol: 36 mint to 90 Min
(xii) 76 ml
Sol: 1L 710ML
= 76 ml : 1710ML
$\Rightarrow \frac{76}{1710}$
= 2: 45
Question 2
There are 2500 students in a school out of them 1100 are girls and the rest are boys , find the ratio of:
Sol: No of girls= 1100
No of boys = 2500- 1100= 1400
(i) Number of boys to number of girls
1400 : 1100 = 14: 11
(ii) Number of girls to number of students
1100: 2500 = 11: 25
(iii) Number of students to number of boys
2500: 1400= 25:14
Question 3
The given figure shows a small garden. The shaded area is reserved for planting flowers and the rest of the area is for grass. Find the ratio of the area of the garden reserved for planting flowers to the area reserved for grass.
(Image to be added)
Sol: Area for flowers = $8 \times 3=24 \mathrm{~m}^{2}$
Area for grass = Total area - area of flowers
$\Rightarrow 12 \times 5-24$
$\Rightarrow \quad 60-24=36 \mathrm{~m}^{2}$
Area of flower : Area of for grass
$24: 36 \Rightarrow \frac{24}{36}=2: 3$
Question 4
If the ratio of the length to the breadth of a rectangle be 5: 3 and its perimeter is 144m, find the length of the rectangle .
Sol: Let the length = 5x & Breadth = 3x
Perimeter of rectangle = $2 \times(l+b)$
$2 \times(5 x+3 x)=144$
$2 \times 8 x=144 \Rightarrow x=\frac{144}{16}=x=9$
Length= 5x = $5 \times 9=45 \mathrm{~m}$ Ans
Question 5
Divide Rs 7400 among three people A, B , C in the ratio 3 : 5: 12
Sol: Let the amount = 3x : 5x : 12x
$321+521+1221=7400$
20x =7400= x = $\frac{7400}{20}$ x = 370
A' s Amount =$3 \times 370$ B's Amount =$5 \times 370$
C's Amount = $12 \times 370$
Question 6
Divide 104 pencils among three children in the Ratio $\frac{1}{2}: \frac{1}{3}: \frac{1}{4}$
Sol: L.C.M of 2,3,4 = 12
Simplified Ratio= $12 \times \frac{1}{2}: 12 \times \frac{1}{3}: 12 \times \frac{1}{4}$
$=6: 4: 3$
Let Sum = $6 x+4 x+3 x=104$
$\Rightarrow \quad 13 x=104$
$\Rightarrow \quad x=\frac{104}{13}$
x= 8
1st share = $6 \times 8=48$ pencils
2nd share = $4 \times 8=32$ pencils
3rd share = $3 \times 8=24$ pencils.
Question 7
The angles of a quadrilateral are in the ratio 1: 2: 3 : 4 . If the sum of the angles of 9 quadrilateral is 360'. Find the measure of each angle.
Sol: Let angles = 1x : 2x : 3x : 4x
1x + 2x+3x + 4x = $360^{\circ} \Rightarrow 10 x=360^{\circ}$
x = $36^{\circ}$
First angle = $1 \times 36=36^{\circ}$
Second angle = $2 \times 36=72^{\circ}$
3rd angle = $3 \times 36=108^{\circ}$
4th angle = $4 \times 36=144^{\circ}$
Question 8
If a: b = 5: 7 and b: c = 14: 15, find a: c
Sol: $\frac{a}{b}=\frac{5}{7} \quad ; \frac{b}{c}=\frac{14}{15}$
$\frac{a}{b} \times \frac{b}{c}=\frac{5}{7} \times \frac{14}{15}$
= $a: c=2: 3$
(ii) If x: y = 2: 7 and y: z = 9: 11 find x: y : z
Sol: $\frac{x}{y}=\frac{2}{7}$ , $\frac{y}{z}=\frac{9}{11}$
L.C.M = 7,9 is 64 : To find x: y: z we have to make y equal
$\frac{x}{y}=\frac{2 \times 9}{7 \times 9}=\frac{18}{63} ; \quad \frac{y}{2}=\frac{9}{11 \times 7} \times \frac{7}{7}=\frac{63}{77}$
$x: y=18: 63 ; \quad y: z=63: 77$
$x: y: z=18: 63: 77$
Question 9
If l: m= $1 \frac{1}{2}: 1 \frac{3}{4}$ and m: n $2 \frac{1}{3}: 4 \frac{1}{5}$
Sol: l: m =$\frac{3}{2}: \frac{7}{4}$= $\frac{3}{2} \div \frac{7}{4}$
=$\frac{3}{2} \times \frac{4}{7}$
$\frac{6}{7}=\frac{l}{m}$
m: n =$\frac{7}{3}: \frac{21}{3}$
$\frac{7}{3} \div \frac{21}{3}=$ $\frac{7}{3} \times \frac{3}{21}$
= $\frac{1}{3}=\frac{m}{n}$
=$\frac{1}{m}=\frac{6}{7}$
=$\frac{m}{n}=\frac{1}{3}$ $\times 7$ $\times 7$ $=\frac{7}{21}$
(i) $\frac{l}{m} \times \frac{m}{n}$
= $\frac{6}{7} \times \frac{7}{21}$
=2: 7= l: n Answer
(ii) l: m: n
Sol: 6: 7: 21
Question 10
Divide Rs 1545 between three people A, B and C such that A gets three fifths of what B gets and the ratio of the share of B to C is 6: 11
Sol: $A=\frac{3}{5} B \Rightarrow \frac{A}{B}=\frac{3}{5} ; \frac{B}{C}=\frac{6}{11}$
L.C.M of 5 and 6 = 30
$\frac{A}{B}=\frac{3 \times 6}{5 \times 6}=\frac{18}{30} ; \quad \frac{B}{C}=\frac{6 \times 5}{11 \times 5}=\frac{30}{55}$
A: B: C = $18: 30: 55$
Sum = $18 x+30 x+55 x=1545$
$\Rightarrow 103x=1545=24=\frac{1545}{103}$
$\Rightarrow x=15$
A' share = $18 \times 15=Rs 270$
B' share = $30 \times 15=Rs 450$
C' share = $55 \times 15$ = Rs 825
Question 11
The ratio of the expenditure to the saving of a family is 7 : 3. Find income if the expenditure is Rs 6300.
Sol: Let Exp. saving= 7x: 3x
Income = Exp. + saving
$=7 x+3 x=10 x$
Given exp = 6300
7x = 6300 = $x=\frac{6300}{7} \Rightarrow 21=900$
Income = 10x = $10 \times 900=9000$ Answer
Question 12
The ratio between the present age of p and Q is 3: 4 respectively . If Q'S Present age is 20
Year what will be the ratio of the ages after 5 years?
Sol: Let ratio of P and Q = 3x: 4x
Q's present age 4x = $20 \Rightarrow 21=\frac{20}{4}=5$
P 's Present age = $3x=3 \times 5=15$
After 5 years p's age = $15+5=20$ yr
Q's age = 20+5 = 25 yr
Ratio P: Q = 20 : 25= 4: 5 answer
Question 13
Rina lost her weight in the ratio 5: 3 . Her original weight was 80 kg . what is her new weight ?
Sol: Original weight = $5 x=80 \Rightarrow x=\frac{80}{5}$
x= 16
New weight = 3x = $3 \times 16=48$kg
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