Question
If , find the value of .
Also: A recipe uses 2 cups of sugar for every 5 cups of flour. How many cups of sugar are needed for 20 cups of flour?
(NCERT Class 6 — fundamental proportion problems)
Solution — Step by Step
When two ratios are equal, they are in proportion. We write:
This means: “3 is to 5 as is to 25.”
Cross-multiplication is the standard technique:
Check: . Simplify by dividing both by 5 → . Correct.
Sugar to flour ratio = .
We need sugar for 20 cups of flour:
Cross-multiply:
We can verify: . Simplify by dividing both by 4 → . Correct.
Why This Works
A proportion says two ratios are equal — they represent the same relationship, just scaled up or down. If 3 out of every 5 parts is a certain quantity, then when we scale up to 25 parts total, the first quantity must scale proportionally to 15.
Cross-multiplication works because if , then multiplying both sides by gives . We’re just clearing the fractions.
Alternative Method — The Multiplier Approach
Instead of cross-multiplication, notice how the second term changed:
means we multiplied by .
So multiply the first term by the same factor: . Done.
For the word problem: means multiplier = . So sugar = .
The multiplier method is faster for simple proportions and helps build number sense. For exams, use whichever method you’re more comfortable with — both give the same answer. The multiplier method is especially handy when the multiplier is a whole number.
Common Mistake
Students sometimes set up the proportion incorrectly. If the question says “3 is to 5 as x is to 25,” make sure 3 and x are in the same position (both numerators or both on the left). Writing would give a wrong answer. The order matters — always match corresponding terms.