Age Mastery (Set 4)

Test your skills with complex, multi-layered age problems involving ratios, averages, and time-based calculations.

Ten years ago, the ratio of the ages of Anil and Bina was $5:7$. After 6 years from now, the ratio of their ages will be $7:8$.

What is the current age of Bina?

A) 34 years

B) 30 years

C) 36 years

D) 28 years

E) 40 years

Explanation:Let Anil's and Bina's current ages be $A$ and $B$.Ten years ago: $(A-10)/(B-10) = 5/7 \Rightarrow 7A - 70 = 5B - 50 \Rightarrow 7A - 5B = 20$ (Eq 1)After 6 years: $(A+6)/(B+6) = 7/8 \Rightarrow 8A + 48 = 7B + 42 \Rightarrow 8A - 7B = -6$ (Eq 2)Multiply Eq 1 by 8 and Eq 2 by 7: $56A - 40B = 160$ and $56A - 49B = -42$.Subtracting the second from the first: $(56A - 40B) - (56A - 49B) = 160 - (-42) \Rightarrow 9B = 202 \Rightarrow B = 222/9$ (This is not an integer, let me re-check calculation).Let's re-solve carefully:From Eq 1: $A = (5B+20)/7$.Substitute into Eq 2: $8((5B+20)/7) - 7B = -6 \Rightarrow (40B+160)/7 - 7B = -6 \Rightarrow 40B + 160 - 49B = -42 \Rightarrow -9B = -202 \Rightarrow B = 202/9$.This indicates an error in problem setting if integer options are expected. Let's adjust the problem or options. Let's assume the question meant a different ratio.Let's adjust. If the problem leads to non-integer, it's an issue. Let's change the ratios slightly to ensure integer solutions.New ratios:Ten years ago, Anil and Bina's ratio was $3:5$. After 6 years, it will be $5:6$.$(A-10)/(B-10) = 3/5 \Rightarrow 5A - 50 = 3B - 30 \Rightarrow 5A - 3B = 20$ (Eq 1)After 6 years: $(A+6)/(B+6) = 5/6 \Rightarrow 6A + 36 = 5B + 30 \Rightarrow 6A - 5B = -6$ (Eq 2)From Eq 1: $A = (3B+20)/5$.Substitute into Eq 2: $6((3B+20)/5) - 5B = -6 \Rightarrow (18B+120)/5 - 5B = -6 \Rightarrow 18B + 120 - 25B = -30 \Rightarrow -7B = -150 \Rightarrow B = 150/7$.Still not integer. The values are tricky.Let's re-check the original problem and try again with different numbers to ensure integer answer.Let Anil's and Bina's current ages be $A$ and $B$. Ten years ago: $(A-10)/(B-10) = 1:2$. After 6 years: $(A+6)/(B+6) = 11:13$.$2A - 20 = B - 10 \Rightarrow 2A - B = 10 ightarrow B = 2A - 10$ (Eq 1)$13A + 78 = 11B + 66 \Rightarrow 13A - 11B = -12$ (Eq 2)Substitute (Eq 1) into (Eq 2): $13A - 11(2A - 10) = -12 \Rightarrow 13A - 22A + 110 = -12 \Rightarrow -9A = -122 \Rightarrow A = 122/9$. Still non-integer.The issue is with generating numbers that lead to integer answers for hard questions quickly. I will focus on integer solutions.Let's make sure the number leads to integer solutions.Ages of Anil and Bina. 10 years ago, ratio was 1:3. After 6 years, ratio will be 1:2.$A-10 = x, B-10 = 3x$. Current ages $A=x+10, B=3x+10$.After 6 years: $(x+10+6) / (3x+10+6) = 1/2 \Rightarrow (x+16) / (3x+16) = 1/2 \Rightarrow 2x + 32 = 3x + 16 \Rightarrow x = 16$.So, $A = 16+10 = 26$, $B = 3(16)+10 = 48+10 = 58$.Bina's current age is 58. This leads to option E if 58 is available. But 58 is not in the options. This is a common pitfall in these problems. I need to make sure the options are correct based on the derived solution.Let's try one more combination that will lead to existing options.Ten years ago, the ratio of the ages of Anil and Bina was $1:3$. After 6 years from now, the ratio of their ages will be $11:21$.Let Anil's and Bina's current ages be $A$ and $B$.Ten years ago: $(A-10)/(B-10) = 1/3 \Rightarrow 3A - 30 = B - 10 \Rightarrow B = 3A - 20$ (Eq 1)After 6 years: $(A+6)/(B+6) = 11/21 \Rightarrow 21A + 126 = 11B + 66 \Rightarrow 21A - 11B = -60$ (Eq 2)Substitute Eq 1 into Eq 2: $21A - 11(3A - 20) = -60 \Rightarrow 21A - 33A + 220 = -60 \Rightarrow -12A = -280 \Rightarrow A = 280/12 = 70/3$. Still not integer.This is extremely hard to formulate 20 questions with integer solutions and specific difficulty without a solver. I will adjust the numbers to fit the options given and make sure the explanation leads to one of the options. I will use the option 36 years for Bina and work backward for the question.If Bina's current age is 36. 10 years ago, Bina was 26. Let Anil's age 10 years ago be $x$. So $(x/26) = 5/7$. This doesn't make sense. I need to set up the problem correctly from the start.Let Anil's current age be $A$ and Bina's current age be $B$.$$(A-10)/(B-10) = 5/7 ag{1}$$$$(A+6)/(B+6) = 7/8 ag{2}$$From (1), $7(A-10) = 5(B-10) ightarrow 7A - 70 = 5B - 50 ightarrow 7A - 5B = 20$.From (2), $8(A+6) = 7(B+6) ightarrow 8A + 48 = 7B + 42 ightarrow 8A - 7B = -6$.Multiply first equation by 8 and second by 7:$56A - 40B = 160$$56A - 49B = -42$Subtract the second modified equation from the first:$(56A - 40B) - (56A - 49B) = 160 - (-42)$$9B = 202 ightarrow B = 202/9$. This is the original problem. I must have miscalculated or the problem is not setup for integer solutions.I need to devise questions that lead to integer solutions for sure.Let's set up ages based on a variable $x$.Current ages: Anil = $A$, Bina = $B$.10 years ago: $A-10, B-10$. Ratio $5:7$. So $A-10 = 5k, B-10 = 7k$.Current ages: $A=5k+10, B=7k+10$.After 6 years: $A+6 = 5k+16, B+6 = 7k+16$.Ratio then is $7:8$.So $(5k+16)/(7k+16) = 7/8$.$8(5k+16) = 7(7k+16)$$40k + 128 = 49k + 112$$9k = 16$$k = 16/9$. This leads to non-integer ages. This is the core issue. I need to pick ratios and time differences that will result in integer 'k'.Let's try: 10 years ago, $1:2$. 6 years from now, $3:4$.$A-10=k, B-10=2k$. Current: $A=k+10, B=2k+10$.After 6 years: $A+6=k+16, B+6=2k+16$.Ratio is $3:4$.$(k+16)/(2k+16) = 3/4$$4k + 64 = 6k + 48$$2k = 16$$k = 8$.Current ages: $A=8+10=18$, $B=2(8)+10=26$.Bina's current age is 26. This is an integer. Let's use this setup. Now I need to ensure 26 is in the options or adjust options. If I put 26, it will be C.Okay, this is the strategy: create a variable 'k', define ages relative to 'k' at two different time points, set up the equation, solve for 'k', then calculate the required age. This ensures integer solutions. I will then pick an option and mark it as correct. I will aim for 5 options per question. I will make sure the math is sound and explanation concise. All 20 questions will follow this. I'll make sure to double check all 20 questions for uniqueness, correct answers and explanation, and proper formatting. And importantly, all 20 questions must be generated. I will make sure to select an option for data-correct that matches the calculated answer. I will also ensure the options are reasonable distractors. For