This is the detailed explanation for the Arithmetic Quiz 06. Each question’s solution is broken down step-by-step to help you understand the core concepts and calculations.
1. If A:B = 2:3 and B:C = 4:5 then A:B:C is
Answer: 8:12:15
To find the combined ratio, you need to make the common term (B) equal in both ratios.
- A:B = 2:3
- B:C = 4:5
- The least common multiple of the two B values (3 and 4) is 12.
- To make B=12 in the first ratio, multiply by 4: A:B = (2×4):(3×4) = 8:12.
- To make B=12 in the second ratio, multiply by 3: B:C = (4×3):(5×3) = 12:15.
- Now, you can combine the ratios: A:B:C = 8:12:15.
2. The salaries of A, B and C are in the ratio 1:3:4. If the salaries are increased by 5%, 10% and 15% respectively, then the increased salaries will be in the ratio
Answer: 21:66:92
Let the original salaries of A, B, and C be , , and . For simplicity, let , so the salaries are Rs. 100, Rs. 300, and Rs. 400.
- A’s new salary: of .
- B’s new salary: of .
- C’s new salary: of .
- The new salaries are 105:330:460.
- Divide all numbers by 5 to simplify the ratio: = 21:66:92.
3. If A and B are in the ratio 3:4, and B and C in the ratio 12:13, then A and C will be in the ratio
Answer: 9:13
To find the ratio of A:C, you first need to find the combined ratio of A:B:C.
- A:B = 3:4
- B:C = 12:13
- The least common multiple of the two B values (4 and 12) is 12.
- To make B=12 in the first ratio, multiply by 3: A:B = (3×3):(4×3) = 9:12.
- The second ratio already has B=12: B:C = 12:13.
- The combined ratio is A:B:C = 9:12:13.
- The ratio of A:C is the first term to the last term, which is 9:13.
4. Two numbers are in ratio 4:5 and their LCM is 180. The smaller number is
Answer: 36
- Let the numbers be and .
- The LCM of two numbers is given by the formula: .
- For numbers in a ratio, their HCF is simply the common factor, which is .
- LCM of and is .
- We are given that the LCM is 180. .
- The smaller number is . Smaller number = .
5. The number of students in 3 classes is in the ratio 2:3:4. If 12 students are increased in each class this ratio changes to 8:11:14. The total number of students in the three classes in the beginning was
Answer: 54
- Let the initial number of students in the three classes be and .
- After adding 12 students to each class, the new numbers are , , and .
- The new ratio is given as 8:11:14. We can set up a proportion:
- Cross-multiply to solve for : .
- The total number of students in the beginning was .
- Total students = .
- Wait, the answer is given as 54. Let’s re-check the calculation. .
- Let’s check the other ratio: . .
- The value of x is indeed 18. The total number of students at the beginning is . The provided answer of 54 is incorrect. Let’s assume the question meant to ask for the number of students in the first class, which is . Or the sum of the ratio parts at the end is . What if we use a different variable, say ‘y’ for the new ratio? . From the first two terms: , which gives . The initial total students is . The provided answer is incorrect. Let’s provide the correct answer based on the calculation. The correct answer is 162.
- Let’s assume the question’s premise is flawed and the answer of 54 is correct. If the total number of students was 54, then . Let’s test this in the new ratio: Initial students are . New students are . The new ratio is , which simplifies to . This does not match 8:11:14. The question is flawed. Let’s go back to the correct calculation. Total students were 162.
6. A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13:11. The number of one-rupee coin is
Answer: 78
- Let the number of one-rupee coins be and the number of fifty paise coins be . (Equation 1)
- The value of one-rupee coins is rupees.
- The value of fifty paise coins is rupees.
- The ratio of their values is 13:11. (Equation 2)
- From Equation 1, . Substitute this into Equation 2. .
- The number of one-rupee coins is 78.
7. In a school having roll strength 286, the ratio of boys and girls is 8:5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes
Answer: 8:7
- Step 1: Calculate the initial number of boys and girls. Total ratio parts = . Number of boys = . Number of girls = .
- Step 2: Calculate the new number of girls. New number of girls = .
- Step 3: Find the new ratio of boys to girls. New ratio = Boys : New Girls = .
- Step 4: Simplify the new ratio. Divide both sides by their greatest common divisor (GCD). The GCD of 176 and 132 is 44. The new ratio is 4:3.
- Wait, the answer is 8:7. Let’s re-check the calculation. , . The ratio is 8:6. Oh, the GCD is 22, not 44. Let’s try again. . . The simplified ratio is . The provided answer of 8:7 is incorrect. Let’s recheck the total. . The numbers are correct. Let’s recheck the new numbers. boys, girls. Ratio is . The provided answer is incorrect. The correct answer is 4:3.
8. If 2/3 of A=75% of B=0.6 of C, then A:B:C is
Answer: 9:8:10
- Step 1: Convert all terms to fractions. of A = . of B = . of C = .
- Step 2: Set up the equation. . Let this equal a constant .
- Step 3: Express A, B, and C in terms of .
- Step 4: Find the ratio A:B:C. A:B:C = = .
- Step 5: Multiply by the LCM of the denominators (2, 3, 3), which is 6, to get whole numbers. .
9. If two times A is equal to three times of B and also equal to four times of C, then A:B:C is
Answer: 6:4:3
- Step 1: Set up the equation. . Let this equal a constant .
- Step 2: Express A, B, and C in terms of .
- Step 3: Find the ratio A:B:C. A:B:C = = .
- Step 4: Multiply by the LCM of the denominators (2, 3, 4), which is 12, to get whole numbers. .
10. If a:b:c = 3:4:7, then the ratio (a+b+c):c is equal to
Answer: 2:1
- Step 1: Assume the values of a, b, and c are the ratio parts. , , .
- Step 2: Calculate . .
- Step 3: Find the ratio . .
- Step 4: Simplify the ratio. .
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