How to Solve Ratio and Proportion Problems
Solve ratio and proportion aptitude problems using the common-multiplier and cross-multiplication methods — with a worked example and practice questions.
Expected Interview Answer
A ratio compares two quantities (a : b), and a proportion states that two ratios are equal (a : b = c : d); most problems are solved by introducing a common multiplier or cross-multiplying.
For a ratio a : b, treat the parts as ax and bx so the total is (a + b)x — this "common multiplier" trick turns shares into solvable equations. A proportion a/b = c/d cross-multiplies to ad = bc, letting you find any missing term. Keep units consistent and reduce ratios to lowest terms. These ideas extend to dividing amounts in a given ratio and to direct/inverse variation.
- The common-multiplier trick solves share problems fast
- Cross-multiplication finds any missing term
- Foundation for mixtures, partnerships and scaling
AI Mentor Explanation
Splitting a partnership’s runs in the ratio 3 : 2 between two batters means thinking of the runs as 3x and 2x, totalling 5x. If together they scored 150, then 5x = 150, so x = 30 and they made 90 and 60. That common-multiplier idea — turning ratio parts into 3x and 2x — is the workhorse of ratio-and-proportion problems, converting a comparison into a solvable equation.
Step-by-Step Explanation
Step 1
Write the ratio
Express parts as ax and bx sharing a common multiplier x.
Step 2
Use the total
Total = (a + b)x; solve for x from the given total.
Step 3
Find each share
Multiply x back into each part (ax, bx).
Step 4
Proportions cross-multiply
For a/b = c/d, use ad = bc to find a missing term.
What Interviewer Expects
- The common-multiplier (ax, bx) technique
- Cross-multiplication for proportions
- Reducing ratios and keeping units consistent
- Applying to shares, mixtures or partnerships
Common Mistakes
- Adding ratio terms without the common multiplier
- Forgetting the total equals (a + b)x
- Mixing inconsistent units
- Not reducing ratios to lowest terms
Best Answer (HR Friendly)
“A ratio compares quantities and a proportion says two ratios are equal. The trick is to treat ratio parts as multiples of a common value — 3x and 2x — use the total to find that value, then compute each share. For proportions, cross-multiply to find any missing term.”
Code Example
def divide_in_ratio(total, a, b):
x = total / (a + b) # common multiplier
return a * x, b * x
print(divide_in_ratio(150, 3, 2)) # (90.0, 60.0)Follow-up Questions
- How do you divide an amount among three people in a ratio?
- What is the difference between direct and inverse proportion?
- How are ratios used in mixture and alligation problems?
- How do you compare two ratios to see which is larger?
MCQ Practice
1. Divide 240 in the ratio 5 : 3. The larger share is?
Total parts = 8, so x = 240 ÷ 8 = 30; the larger share is 5 × 30 = 150.
2. In the proportion 4 : 6 = 6 : x, x equals?
Cross-multiply: 4x = 36, so x = 9.
3. The ratio 12 : 18 in lowest terms is?
Divide both by 6 → 2 : 3.
Flash Cards
Common-multiplier trick? — Write ratio parts as ax and bx; total = (a + b)x.
Proportion rule? — a/b = c/d ⇒ ad = bc (cross-multiply).
Divide N in ratio a : b? — x = N/(a+b); shares are ax and bx.
Always do what first? — Reduce ratios to lowest terms and keep units consistent.