How do you solve ratio and age problems?

A ratio tells you the size of the shares, not the actual amounts. If three friends split a Rs.1,200 Swiggy bill in the ratio 2 : 3 : 5, think of the bill as 2 + 3 + 5 = 10 equal parts. Each part is Rs.120, so they pay Rs.240, Rs.360 and Rs.600.

Age problems are ratio problems with time added. The key fact: everyone gets older at the same speed. In 6 years, both you and your elder brother are 6 years older, so the gap between your ages stays exactly the same forever. The ratio, however, keeps changing.

Question 1: The ratio of Aman's age to Babita's age is 4 : 5. After 6 years it will be 5 : 6. Find their present ages.

1. Let the present ages be 4x and 5x.
2. After 6 years: (4x + 6) / (5x + 6) = 5 / 6.
3. Cross-multiply: 6(4x + 6) = 5(5x + 6), so 24x + 36 = 25x + 30.
4. x = 6. Aman is 24 and Babita is 30.
5. Check: in 6 years they are 30 and 36, and 30 : 36 = 5 : 6.

Question 2: The sum of a father's and son's ages is 50. Five years ago, the father was 7 times as old as the son. Find their ages.

1. Let the son be S. The father is 50 - S.
2. Five years ago: (50 - S - 5) = 7(S - 5), so 45 - S = 7S - 35.
3. 80 = 8S, so S = 10 and the father is 40.
4. Check: five years ago they were 5 and 35, and 35 = 7 x 5.

let x = 1;
while ((4 * x + 6) * 6 !== (5 * x + 6) * 5) x++;
console.log([4 * x, 5 * x]); // Output: [24,30]

• Ratio a : b means the values are ax and bx for some x. Find x from one more piece of information.
• Dividing an amount in a ratio: each share = total x (its part) / (sum of parts).
• Combining ratios: if A : B = 2 : 3 and B : C = 4 : 5, make B the same in both. A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
• Ages: the difference between two people's ages is constant. "n years ago" means subtract n from each person; "after n years" means add n.
• Option checking: in exams, plugging the options into the condition is often faster than solving.

Ratios show up everywhere: splitting rent among flatmates, mixing cement and sand, a recipe scaled for 20 guests at a college fest, or the boy-to-girl ratio in a batch. Age and ratio questions are standard in TCS NQT, Capgemini, Accenture, SSC and bank exams.

• Adding years to only one person.
• Writing the ages as 4 and 5 instead of 4x and 5x.
• Mixing up the order in the ratio and getting the two people swapped.
• Assuming the ratio stays the same over time. Only the difference stays the same.
• Not checking the answer against both conditions.

• A is twice as old as B. Ten years ago, A was three times as old as B. Find their present ages.
• Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.
• If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.