Ratio
A ratio is used to compare two or more quantities of the same kind which are measured in the same unit.
The idea of ratio is very important and occurs in many branches of mathematics.
A ratio is usually expressed as a fraction in its lowest terms.
The ratio of a is to b, where a and b represent two quantities of the same kind, and b not = 0, is written as a:b.
Word Problem:
1. There are 17 boys and 19 girls in a class. Find the ratio of
i) The number of boys to the number of girls,
ii) The number of girls to the number of boys.
Solution:
Ratio of the number of boys to the number of girls = 17 : 19
Ratio of the number of girls to the number of boys = 19 : 17
1:2, 2:4 and 6 : 12 are known as equivalent ratios.
Q.2. If x : y = 2:3, find an equation connecting x and y.
x/y = 2/3
3x = 2y
x = 2/3 y
3. A farmer owns 50 head of cattle and 225 sheep. Find the ratio of the number of cattle to the number of sheep. Find also the ratio of the number of sheep to the total number of cattle and sheep.
Solution:
Cattle = 50
Sheep = 225
Cattle : Sheep = 50 : 225
= 50/225
= 2/9 Ans.
Total number of Cattle and Sheep = 50 + 225
= 275
Sheep : total Cattle & Sheep
= 225 : 275
= 225/275
= 9/11 Ans.
4. If each side of a square is enlarged three times, by how many times is its area increased?
Solution:
Suppose each side of the square is = a
It's area = a2
If each side is enlarged 3 times, then each side will be = 3a
So the area of the new square
= (3a)2
= 9a2
So, the ratio = 9a2: a2
= 9a2/a2
= 9 /1
= 9:1 Ans.
5. Express the ratio 2.5 : 8.0 in the form 1:n.
Solution: 2.5 : 8.0
= 2.5/8.0
= 25/80
= 5/16
= 1 : 3.2 Ans.
6. A map is drawn on a scale of 2 cm to 1 km. Express its representative fraction in the form 1:n.
Solution:
The ratio is
2 cm : 1km
= 2cm / 1km
= 2cm / 100000 cm
= 1/ 50000
= 1 : 50000 Ans.
7. In a village there are 440 men and 250 women. Find the ratio of the number of men to number of women. Find also the ratio of the number of men to the total number of men and women, and the ratio of the number of women to the total number of men and women.
men : women = 440 : 250
= 44 : 25
Total number of men and women = ( 440 + 250)
= 690
men : total number of men & women = 44 : 69
women : total number of men & women = 25 : 69
8. I have two square lawns. The first is 20 metres square and the second 35 metres square. Find the ratio of the times it takes me to cut the lawns.
Solution:
Area of the first square = (20)²
= 400 m²
Area of the second square = (35)²
= 1225 m²
So, the ratio = 400 m² : 1225m²
= 16: 49 Ans.
9. A cube has each of its edges doubled. Find the ratio of the volume of the first cube to that of the second. Find also the ratio of the area of the face of the first cube to that of the second.
Solution:
Suppose, each edge of the cube = a
So, the volume is = a³
Each edge of the new cube = 2a
So, the volume is = (2a)³
= 8a³
Therfore, the ratio of the volume of the first cube tio that of the second
= a³ : 8a³
= a³/8a³
= 1: 8
Area of the face of the first cube = a.a
= a²
And area of the face of the second cube = 2a . 2a
= 4a²
Therefore, the ratio of the face of the first cube to the second
= a² : 4a²
= 1:4 Ans.
10. After driving 30 km, a motorist estimates that he has covered 18 km of good road and 12 km of poor road. Find the ratio of good road to poor road and the ratio of good road to the total distance covered.
Solution:
good road = 18 km
poor road = 12 km
Good road : Poor road = 18 km : 12km
= 18km/12km
= 3: 2 Ans.
Good road : total distance = 18 km : 30km
= 18 km /30km
= 3/5
= 3 : 5 Ans.
11. To catch a train, a man has to go a distance of 4 km. He walks some of the way but has to run a certain distance. The ratio of the distance walked to the distance run is 5: 3. How far does the man run? Find the ratio of the distance walked to the whole distance.
Solution:
Distance walked : distance run = 5 : 3
Sum of the ratio = 5 + 3
= 8
Distance run = 4 x 3/8
= 1.5 km
Distance walked = 4 x 5/8
= 2.5 km
Distance walked : total distance
= 2.5 km : 4 km
= 5/8 Ans.
12. The width of a pavement is 1.05 m and the width of a road 6m. Find the ratio of the width of pavement to the width of road, assuming the road is paved on each side.
Solution:
Width of the pavement = 1.05 m
Width of the road = 6m
The road is paved on each side
So, the total width of the pavement = 1.05 x 2m
= 2.10 m
Width of the pavement : width of the road
= 2.1 m : 6m
= 7/20 Ans.
9.1 Ratio
Q.1. There are 17 boys and 19 girls in a class. Find the ratio of
i) the number of boys to the number of girls,
ii) the number of girls to the number of boys
Solution:
i) Ratio of the number of boys to the number of girls = 17 : 19
ii) Ratio of the number of girls to the number of boys = 19 : 17
Q.2. There are 33 lemons and 20 pears in a basket. Find the ratio of
i) the number of lemons to the number of pears
ii) the number of pears to the total number of fruits in the basket.
Solution:
i) the number of lemons to the number of pears = 33 : 20
ii) the number of pears to the total number of fruits in the basket = 20: 53
Expressing Ratios as Fractions:
Q.3. The ratio of the number of female participants to the number of male participants at a party is 4: 9. If there are 30 more male participants than female participants, calculate the total number of people who attended the party.
Q.4. The ratio of the number of fiction of fiction books to the number of non-fiction books in a library is 5 : 2. If there are 1421 fiction and non-fiction books altogether, how many more fiction than non-fiction books are there in the library?
Ratio involving three Quantities:
Problem involving Ratios of Three Quantities:
Q.2. A sum of money is divided among Khairul, Michael and Ethan in the ratio 6:4:5. After Khairul gives $30 to Michael and $15 to Ethan, the ratio becomes 7:6:7. Find the amount of money Khairul had initially. Ans: $360
Ex.
Basic Level:
https://forms.gle/mzB4cZ5JnV7qaSft9
https://docs.google.com/forms/d/1BXV4GSNBSKGnpuC6rr50vXCAbUj5RrZA-udLmE6mcAg/edit?usp=sharing