Problem 1 :
The sum of two numbers is 100. If the ratio between the two numbers is 2 : 3, find the numbers.
Problem 2 :
If the angles of a triangle are in the ratio 2 : 7 : 11, then find the angles.
Problem 3 :
The ratio of two numbers is 7 : 10. Their difference is 105. Find the numbers.
Problem 4 :
A, B and C are three cities. The ratio of average temperature between A and B is 11:12 and that between A and C is 9 : 8. Find the ratio between the average temperature of B and C.
Problem 5 :
The ratio between the speeds of two trains is 7 : 8. If the second train runs 400 miles in 5 hours, then, find the speed of the first train.
Problem 6 :
Find in what ratio will the total wages of the workers of a factory be increased or decreased, if there be a reduction in the number of workers in the ratio 15:11 and an increment in their wages in the ratio 22 : 25.
Problem 7 :
If $782 is divided among three persons A, B and C in the ratio 1/2 : 2/3 : 3/4, then find the share of A.
Problem 8 :
An amount of money is to be divided among P, Q and R in the ratio 3 : 7 : 12. The difference between the shares of P and Q is $2400. What will be the difference between the shares of Q and R?
1. Answer :
From the given ratio 2 : 3,
first number = 2x
second number = 3x
Given : The sum of the two numbers is 100.
2x + 3x = 100
5x = 100
Divide both sides by 5
x = 20
Therefore,
first number = 2(20) = 40
second number = 3(20) = 60
2. Answer :
From the ratio 2 : 7 : 11, the three angles can be assumed to be
2x, 7x, 11x
In any triangle, sum of the angles is equal to 180°.
2x + 7x + 11x = 180°
20x = 180
x = 9
Then,
the first angle = 2x = 2 ⋅ 9 = 18°
the second angle = 7x = 7 ⋅ 9 = 63°
the third angle = 11x = 11 ⋅ 9 = 99°
Hence the angles of the triangle are (18°, 63°, 99°).
3. Answer :
From the given ratio 7 : 10, the two number can be assumed to be
7x, 10x
The difference between the numbers is 105.
10x - 7x = 105
3x = 105
x = 35
Then,
the first number = 7x = 7 ⋅ 35 = 245
the second number = 10x = 10 ⋅ 35 = 350
Hence the numbers are 245 and 350.
4. Answer :
From A : B = 11 : 12 and A : C = 9 : 8, we find A in common.
The values corresponding to A in both the ratios are different.
First we have to make them to be same.
value corresponding to A in the 1^{st} ratio = 11
value corresponding to A in the 2^{nd} ratio = 9
L.C.M of (11, 9) = 99.
First ratio ----> A : B = 11 : 12 = (11 ⋅ 9) : (12 ⋅ 9) = 99 : 108
Second ratio ----> A : C = 9 : 8 = (9 ⋅ 11) : (8 ⋅ 11) = 99 : 88
Clearly,
A : B = 99 : 108 ----(1)
A : C = 99 : 88 ----(2)
Now, the values corresponding to A in both the ratios are same.
From (1) and (2), we get
B : C = 108 : 88
Simplify.
B : C = 27 : 22
Hence, the ratio between the average temperature of B and C is
27 : 22
5. Answer :
From the given ratio 7 : 8, we have
speed of the first train = 7x ----(1)
speed of the second train = 8x ----(2)
Second train runs 400 miles in 5 hours (given)
[Hint : Speed = Distance/Time]
So, speed of the second train is
= 400/5
= 80 mph ----(3)
From (2) and (3), we get
8x = 80
x = 10
From (1), speed of the first train is
= 7x
= 7 ⋅ 10
= 70
Hence, the speed of the first train is 70 mph.
6. Answer :
Let x be the number of workers and y be the average wages per worker.
Then, total wages = (no. of workers) x (wages per worker)
Total wages = xy or 1xy ----(1)
After reduction in workers in the ratio 15 : 11,
no. of workers = 11x/15
After increment in wages in the ratio 22 : 25,
wages per worker = 25y/22
Now, the total wages is
= (11x/15) ⋅ (25y/15)
= 5xy/6
So, the total wages after changes is
= 5xy/6 -----(2)
From (1) and (2) we get that the total wages get decreased from xy to 5xy/6.
So, the decrement ratio is
= xy : 5xy/6
Divide both the terms by xy.
= 1 : 5/6
Multiply both the terms by 6.
= 6 : 5
Hence, the total wages will be decreased in the ratio
6 : 5
7. Answer :
Given ratio ---> 1/2 : 2/3 : 3/4.
First let us convert the terms of the ratio into integers.
L.C.M of denominators (2, 3, 4) = 12.
When we multiply each term of the ratio by 12, we get
12 ⋅ 1/2 : 12 ⋅ 2/3 : 12 ⋅ 3/4 ----> 6 : 8 : 9
From the ratio 6 : 8 : 9,
share of A = 6x
share of B = 8x
share of C = 9x
We know that,
share of A + share of B + share of C = 782
6x + 8x + 9x = 782
23x = 782
x = 34
Share of A is
= 6x
= 6 ⋅ 34
= 204
So, the share of A is $204.
8. Answer :
From the given ratio 3 : 7 : 12,
share of P = 3x
share of Q = 7x
share of R = 12x
Difference between the shares of P and Q is $ 2400.
share of Q - share of P = 2400
7x - 3x = 2400
4x = 2400
x = 600
Difference between the shares of Q and R is
= 12x - 7x
= 5x
= 5 ⋅ 600
= 3000
So, the difference between the shares of Q and R is $3000.
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