All questions of Ratio & Proportion for Class 10 Exam

Given information:
- Age of father = A
- Age of mother = B
- Age of son = C
- Age of grandson = D
- A - B = 3
- B * C = 78
- C * D = 33
- Average age of the family = 34 years

To find: B - C

Step 1: Finding the values of A, B, C, and D
We are given that the average age of the family is 34 years. Therefore, the sum of their ages is 34 * 4 = 136 years.

We know that A - B = 3, so A = B + 3.

From the given information, we can also deduce that D = C * (33/C) = 33.

Now, substituting the values of A and D in the sum of their ages equation, we get:
A + B + C + D = (B + 3) + B + C + 33 = 136

Simplifying the equation, we have:
2B + 2C + 36 = 136
2B + 2C = 100
B + C = 50

Step 2: Finding B - C
We need to find the value of B - C.

Substituting the value of B + C from the previous step, we have:
B - C = (B + C) - 2C = 50 - 2C

To solve this equation, we need to find the value of C.

From the given information, we know that B * C = 78. We can find the factors of 78 to determine the possible values of B and C.

The factors of 78 are: 1, 2, 3, 6, 13, 26, 39, 78.

Since B + C = 50, the possible values of B and C are:
(1, 49), (2, 48), (3, 47), (6, 44), (13, 37), (26, 24)

Out of these possibilities, we need to find the value of C that satisfies the equation B * C = 78.

The only value that satisfies this equation is B = 13 and C = 37.

Substituting these values in B - C, we have:
B - C = 13 - 37 = -24

However, the age cannot be negative. Therefore, we need to find the absolute value of B - C, which is 24.

Thus, the correct answer is option D) 22.

To find the ratio a : b : c : d : e, we need to find the values of a, b, c, d, and e. Let's solve the problem step by step.

Given ratios:
a : b = 3 : 2
b : c = 2 : 1
c : d = 1/3 : 1/7
d : e = 1/4 : 1/5

Step 1: Finding the value of b
From the first ratio, a : b = 3 : 2, we can deduce that b is 2/3 of a. To find the value of b, we can assume a constant value for a, let's say 3. Therefore, b = (2/3) * 3 = 2.

Step 2: Finding the value of c
From the second ratio, b : c = 2 : 1, we can deduce that c is half of b. Since b is 2, c = 2/2 = 1.

Step 3: Finding the value of d
From the third ratio, c : d = 1/3 : 1/7, we can write it as c : d = 7/21 : 3/21. This implies that d is 3/21 of c. Since c is 1, d = (3/21) * 1 = 3/21 = 1/7.

Step 4: Finding the value of e
From the fourth ratio, d : e = 1/4 : 1/5, we can write it as d : e = 5/20 : 4/20. This implies that e is 4/20 of d. Since d is 1/7, e = (4/20) * (1/7) = 4/140 = 1/35.

Step 5: Writing the final ratio
Now that we have the values of a, b, c, d, and e, we can write the final ratio:
a : b : c : d : e = 3 : 2 : 1 : 1/7 : 1/35

To simplify this ratio, we need to find a common denominator. The common denominator is 35. Multiplying all the terms by 35, we get:
a : b : c : d : e = 105 : 70 : 35 : 5 : 1

Since 5 : 1 is equivalent to 15 : 3, we can rewrite the ratio as:
a : b : c : d : e = 105 : 70 : 35 : 15 : 3

Simplifying further, we can divide all the terms by 5, resulting in:
a : b : c : d : e = 21 : 14 : 7 : 3 : 1

Therefore, the correct answer is option 'C': 105 : 70 : 35 : 15 : 3.

Understanding the Ratios
To solve the problem, we need to interpret the ratios given for A, B, and C.
- A : B = 5 : 2
- B : C = 7 : 13
Finding a Common Ratio
1. Express B in terms of A:
- From A : B = 5 : 2, we can express B as:
- B = (2/5)A
2. Express C in terms of B:
- From B : C = 7 : 13, we can express C as:
- C = (13/7)B
3. Substituting B:
- Replace B in the equation for C:
- C = (13/7) * (2/5)A
- C = (26/35)A
Setting Up the Total
Now, we have:
- A = A
- B = (2/5)A
- C = (26/35)A
Summing the Shares
The total amount can be expressed as:
- Total = A + B + C
- Total = A + (2/5)A + (26/35)A
To combine these, we first need a common denominator, which is 35:
- A = (35/35)A
- B = (14/35)A
- C = (26/35)A
Now, summing these gives:
- Total = (35/35 + 14/35 + 26/35)A = (75/35)A = (15/7)A
Calculating A's Share
We know that the total amount is Rs. 750. Therefore:
- (15/7)A = 750
To find A:
A = 750 * (7/15) = 350
Conclusion
Thus, A’s share is:
- Rs. 350
Hence, the correct answer is option 'B'.

To solve this problem, we need to set up a system of equations based on the given information. Let's assume that the number of one-rupee coins is x and the number of fifty paise coins is y.

1. Setting up the equations:
We are given that the ratio of the values of one-rupee coins to fifty paise coins is 13:11. This means that the value of one-rupee coins is 13 times the value of fifty paise coins.

Therefore, we can write the equation:
1 * x = 13 * 0.50 * y

We are also given that the total number of coins is 210. So we can write another equation:
x + y = 210

2. Solving the equations:
First, let's simplify the first equation:
x = 6.50y

Now substitute this value of x into the second equation:
6.50y + y = 210
7.50y = 210
y = 210 / 7.50
y = 28

Now substitute the value of y back into the second equation to find x:
x + 28 = 210
x = 210 - 28
x = 182

So, the number of one-rupee coins (x) is 182.

3. Finding the correct answer:
We need to find the number of one-rupee coins, which is 182. Among the given options, option D states that the number of one-rupee coins is 78. This is incorrect.

Therefore, the correct answer is option D. The number of one-rupee coins is actually 182.