(a) What is the ratio of number of girls to the number of boys?
(b) What is the ratio of number of girls to the total number of students in the class?
Solutions:
Number of girls = 20 girls
Number of boys = 15 boys
The total number of students = 20 + 15 = 35
(a) The ratio of \(\text{the number of girls}\over \text{number of boys}\) = \({20 \over {15}} ={ 4\over3 }\)
(b) The ratio of \(\text{the number of girls}\over \text{ total number of students }\)= \({20 \over {35}} = {4 \over 7}\)
2. Out of 30 students in a class, 6 like football,
12 like cricket and remaining like tennis. Find the ratio of
(a) Number of students liking football to
number of students liking tennis.
(b) Number of students liking cricket to total number of students.
Solutions:
The number of students who like football = 6
The number of students who like cricket = 12
The number of students who like tennis = 30 – 6 – 12 = 12
(a) Ratio of \(\text{the number of students liking football }\over \text{the number of students liking tennis}\) =\({6 \over{ 12}} = {1\over 2} \)
(b) \(\text{The number of students liking cricket }\over \text{ total number of students}\) = \({{12} \over{30}}= {2\over 5} \)
3. See the figure and find the ratio of
(a) Number of triangles to the number of circles inside the rectangle.
(b) Number of squares to all the figures inside the rectangle.
(c) Number of circles to all the figures inside the rectangle.
Solutions:
Given in the figure
The number of triangles = 3
The number of circles = 2
The number of squares = 2
The total number of figures = 7
(a) The ratio of \(\text{the number of triangles}\over \text{ the number of circles inside the rectangle} \) = \({3\over 2 }\)
(b) The ratio of \(\text{the number of squares } \over \text{ all the figures inside the rectangle} \) = \( {2 \over 7 }\)
(c) The ratio of \(\text{the number of circles} \over \text{ all the figures inside the rectangle} \) = \( {2 \over 7 }\)
4. Distances travelled by Hamid and Akhtar in an hour are 9 km and 12 km. Find the ratio of speed of Hamid to the speed of Akhtar.
Solutions:
Distance travelled by Hamid in one hour = 9 km
Distance travelled by Akhtar in one hour = 12 km
Speed of Hamid = 9 km/hr
Speed of Akhtar = 12 km/hr
The ratio of \(\text{ the speed of Hamid} \over \text{ the speed of Akhtar } \) = \( 9 \over 12\) = \(3 \over 4\)
5. Fill in the following blanks:
\(15 \over 18\)=\({\boxed{}} \over 6\)=\(10 \over {\boxed{}}\)=\({\boxed{}} \over 30\)[Are these equivalent ratios?]
Solutions:
\({15 \over 18} = {(5 × 3) \over (6 × 3)} ={5 \over 6}\)
\({10 \over 12}={(5 × 2) \over (6 × 2) } ={5 \over 6}\)
\({25 \over 30} ={(5 × 5) \over (6 × 5) } ={5 \over 6}\)
Hence, 5, 12 and 25 are the numbers which come in the blanks, respectively.
Yes, all are equivalent ratios.
6. Find the ratio of the following :
(a) 81 to 108
(b) 98 to 63
(c) 33 km to 121 km
(d) 30 minutes to 45 minutes
Solutions:
(a)\({81 \over 108} = {(3 × 3 × 3 × 3) \over (2 × 2 × 3 × 3 × 3)} ={3 \over 4}\)
(b)\({98 \over 63}={(14 × 7) \over (9 × 7) } ={14 \over 9}\)
(c)\({33 \over 121} ={(3 × 11) \over (11 × 11) } ={3 \over 11}\)
(d)\({30 \over 45} ={(2 × 3 × 5) \over (3 × 3 × 9) } ={2 \over 3}\)
7. Find the ratio of the following:
(a) 30 minutes to 1.5 hours
(b) 40 cm to 1.5 m
(c) 55 paise to ₹ 1
(d) 500 mL to 2 litres
Solutions:
(a) 30 minutes to 1.5 hours
30 min = \({30 \over 60} = 0.5\) hours
Required ratio = \({(0.5 × 1) \over (0.5 × 3)}
= {1 \over 3}\)
(b) 40 cm to 1.5 m
1.5 m = 150 cm
Required ratio = \({40 \over 150} = {4 \over 15} \)
(c) 55 paise to ₹ 1
₹ 1 = 100 paise
Required ratio = \({55 \over 100} ={(11 × 5) \over (20 × 5)} = {11 \over 20} \)
(d) 500 ml to 2 litres
1 litre = 1000 ml
2 litre = 2000 ml
Required ratio = \({500 \over 2000} = {5 \over 20 }= {5 \over (5 × 4) }= {1\over 4}\)
8. In a year, Seema earns ₹ 1,50,000 and saves ₹ 50,000. Find the ratio of
(a) Money that Seema earns to the money she saves.
(b) Money that she saves to the money she spends.
Solutions:
Money earned by Seema = ₹ 150000
Money saved by Seema = ₹ 50000
Money spent by Seema = ₹ 150000 – ₹ 50000 = ₹ 100000
(a) The ratio of the money earned to money saved = \(\frac{150000 }{ 50000} = {15 \over 5} = {3 \over 1} \)
(b) The ratio of the money saved to money spent = \(\frac{50000 }{ 100000} = {5 \over 10} = {1 \over 2} \)
9. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students.
Solutions:
The number of teachers in a school = 102
The number of students in a school = 3300
The ratio of the number of teachers to the number of students = \({102 \over 3300} = {(2 × 3 × 17)\over (2 × 3 × 550)} = {17 \over 550}\)
10. In a college, out of 4320 students, 2300 are girls. Find the ratio of
(a) Number of girls to the total number of students.
(b) Number of boys to the number of girls.
(c) Number of boys to the total number of students.
Solutions:
The total number of students = 4320
The number of girls = 2300
The number of boys = 4320 – 2300
= 2020
(a) The ratio of the number of girls to the total number of students = \({2300 \over 4320} = {(2 × 2 × 5 × 115) \over (2 × 2 × 5 × 216)} = {115 \over 216 }\)
(b) The ratio of the number of boys to the number of girls = \({2020 \over 2300} = {(2 × 2 × 5 × 101) \over (2 × 2 × 5 × 115)} = {101 \over 115 }\)
(c) The ratio of the number of boys to the total number of students = \({2020 \over 4320} = {(2 × 2 × 5 × 101) \over (2 × 2 × 5 × 216)} = {101 \over 216 }\)
11. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and remaining
opted table tennis. If a student can opt only one game, find the ratio of
(a) Number of students who opted basketball to the number of students who opted
table tennis.
(b) Number of students who opted cricket to the number of students opting basketball.
c) Number of students who opted basketball to the total number of students.
Solutions:
(a) The ratio of the number of students who opted basketball to the number of students who opted table tennis =\({ 750 \over 250} = {3 \over 1}\)
(b) The ratio of the number of students who opted cricket to the number of students opting basketball
= \({800 \over 750} = {16 \over 15}\)
(c) The ratio of the number of students who opted basketball to the total number of students
= \({750 \over 1800 }= {25 \over 60} = {5 \over 12 }\)
12. Cost of a dozen pens is ₹180 and cost of 8 ball pens is ₹ 56. Find the ratio of the cost of a pen to the cost of a ball pen.
Solutions:
The cost of a dozen pens = ₹ 180
The cost of 1 pen = \({180 \over 12} \) = ₹ 15
The cost of 8 ball pens = ₹ 56
The cost of 1 ball pen = \({56 \over 8}\) = ₹ 7
Hence, the required ratio is \(15 \over 7\).
13. Consider the statement: Ratio of breadth and length of a hall is 2 : 5.
| The breadth of the hall (in meters) | 10 | … | 40 |
| The length of the hall (in meters) | 25 | 50 | … |
Complete the
following table that shows some possible breadths and lengths of the hall.
Solutions:
(i) Length = 50 m
\({ \text{Breadth} \over 50} = {2 \over 5} \)
By cross multiplication,
5× Breadth = 50 × 2
Breadth = \({(50 × 2)\over 5} = {100 \over 5} = 20\) m
(ii) Breadth = 40 m
\({40 \over \text{Length}} = {2 \over 5}\)
By cross multiplication,
\(2 × \text{Length} = 40 × 5 \)
Length =\({(40 × 5) \over 2}\)
Length = \(200 \over 2\)
Length = 100 m
14. Divide 20 pens between Sheela and Sangeeta in the ratio of 3 : 2.
Solutions:
Terms of 3: 2 = 3 and 2
The sum of these terms = 3 + 2
= 5
Now, Sheela will get \(3 \over 5\) of the total pens, and
Sangeeta will get \(2 \over 5\) of the total pens.
The number of pens Sheela has = 3 / 5 × 20
= 3 × 4
= 12
The number of pens Sangeeta has = \({2 \over 5} × 20
= 2 × 4
= 8 \)
15. Mother wants to divide ₹ 36 between her daughters Shreya and Bhoomika in the ratio
of their ages. If age of Shreya is 15 years and age of Bhoomika is 12 years, find how much Shreya and Bhoomika
will get.
Solutions:
Ratio of ages = \(15 \over 12\) = 5 / 4
Hence, the mother wants to divide ₹ 36 in the ratio of 5: 4.
Terms of 5: 4 are 5 and 4
The sum of these terms = 5 + 4
= 9
Here, Shreya will get \(5 \over 9\) of the total money, and Bhoomika will get \(4 \over 9\) of the total money.
The amount Shreya gets = \(5 \over 9\) × 36
= ₹20
The amount Bhoomika gets = \(4 \over 9\) × 36
= ₹16
Therefore, Shreya will get ₹ 20, and Sangeeta will get ₹ 16.
16. Present age of father is 42 years and that of his son is 14 years. Find the ratio of
(a) Present age of father to the present age of son.
(b) Age of the father to the age of son, when son was 12 years old.
(c) Age of father after 10 years to the age of son after 10 years.
(d) Age of father to the age of son when father was 30 years old.
Solutions:
(a) Present age of father = 42 years
Present age of son = 14 years
Required ratio \({42 \over 14} = {3\over 1}\)
(b) The son was 12 years old 2 years ago.
So, the age of the father 2 years ago will be
= 42 – 2 = 40 years
Required ratio = \({40 \over 12} = {(4 × 10) \over (4 × 3)} = {10 \over 3}\)
(c) After ten years age of the father = 42 + 10 = 52 years
After 10 years age of the son = 14 + 10 = 24 years
Required ratio = \({52 \over 24} = {(4 × 13)\over (4 × 6)} = {13 \over 6 }\)
(d) 12 years ago, age of the father was 30.
At that time, the age of the son = 14 – 12
= 2 years
Required ratio = \({30 \over 2} = {(2 × 15) \over 2} = {15 \over 1}\)
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