If 40% of a number is 80 find 65% of the same number

Question

If 40% of a number is 80, find 65% of the same number.

Solution — Step by Step

Let the unknown number be $x$ . We’re told 40% of $x$ is 80. Writing this as an equation:

\frac{40}{100} \times x = 80

Multiply both sides by $\frac{100}{40}$ :

x = 80 \times \frac{100}{40} = 80 \times 2.5 = 200

The original number is 200.

65\% \text{ of } 200 = \frac{65}{100} \times 200 = \frac{65 \times 200}{100} = \frac{13000}{100} = 130

Why This Works

The key insight is that “percent” literally means “per hundred.” So 40% = 40/100. Setting up the equation $\frac{40}{100} \times x = 80$ translates the English sentence directly into algebra.

Once we have the original number, finding any other percentage is straightforward — multiply by that fraction.

Alternative Method

We can skip finding the number entirely using proportional reasoning.

If 40% = 80, then 1% = 80 ÷ 40 = 2.

So 65% = 65 × 2 = 130.

The “find 1% first” shortcut is the fastest method. Once you know 1% of the number, multiply by any percentage you want. Here: 1% = 2, so 65% = 130, 80% = 160, 25% = 50, etc. — all in one step.

Common Mistake

A very common error: students write $40 \times 80 / 100 = 32$ and think that’s the number. This confuses the problem — 40% of 80 is not what we want. We want the number WHOSE 40% IS 80. Always re-read: “40% of [unknown] = 80” not “40% of 80.”

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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