Arithmetic

Profit & Loss

Medium

Identical chocolate pieces are sold in boxes of two sizes, small and large. The large box is sold for twice the price of the small box. If the selling price per gram of chocolate in the large box is $12 \%$ less than that in the small box, then the percentage by which the weight of chocolate in the large box exceeds that in the small box is nearest to

$127$

$124$

$144$

$135$

✅ Correct Option: 1

Let us break this problem down, making it crystal clear for you to understand.

What we know:

Large box costs twice as much as small box

Price per gram in large box is 12% less than in small box

We need to find: by what percentage does the weight in large box exceed the small box?

Let us define our variables:

Small box price = P

Large box price = 2P (given: twice the price)

Weight in small box = W₁

Weight in large box = W₂

Price per gram in small box = Total price ÷ Total weight = $\dfrac{P}{W_1}$

Price per gram in large box = $\dfrac{2P}{W_2}$

Here is the key insight: The price per gram in the large box is 12% less than in the small box.

What does "12% less" mean?

If something is 12% less, it means it is 88% of the original value.

(100% - 12% = 88% = 0.88)

So: Price per gram in large box = 0.88 × (Price per gram in small box)

Setting up the equation:

$\dfrac{2P}{W_2} = 0.88 \times \dfrac{P}{W_1}$

$\dfrac{2P}{W_2} = \dfrac{0.88P}{W_1}$

Divide both sides by P:

$\dfrac{2}{W_2} = \dfrac{0.88}{W_1}$

Cross multiply:

$2 \times W_1 = 0.88 \times W_2$

Solve for $\dfrac{W_2}{W_1}$ :

$\begin{aligned} \dfrac{W_2}{W_1} &= \dfrac{2}{0.88} \\ &= \dfrac{200}{88} \\ &= \dfrac{25}{11} \end{aligned}$

Formula for percentage increase:

Percentage increase = $\dfrac{\text{New value - Old value}}{\text{Old value}} \times 100$

In our case:

Percentage increase = $\dfrac{W_2 - W_1}{W_1} \times 100$

We can rewrite this as:

$= \left(\dfrac{W_2}{W_1} - 1\right) \times 100$

$= \left(\dfrac{25}{11} - 1\right) \times 100$

$= \left(\dfrac{25}{11} - \dfrac{11}{11}\right) \times 100$

$= \dfrac{14}{11} \times 100$

$= 127.27\%$

The weight of chocolate in the large box exceeds that in the small box by approximately 127%.

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