Understanding the Fraction Calculation Problem

This problem involves calculating a fraction of a number. Amit made a mistake by using the wrong fraction, which resulted in an answer that was greater than the correct one. We need to find the original number.

Setting Up the Equation to Find the Number

Let the unknown number be denoted by \(x\).
According to the problem, Amit calculated \(\frac{2}{5}\) of the number. This incorrect calculation is given by \(\frac{2}{5}x\).
The correct calculation should have been \(\frac{2}{15}\) of the number. This correct calculation is given by \(\frac{2}{15}x\).
The problem states that Amit's answer (\(\frac{2}{5}x\)) was greater than the correct answer (\(\frac{2}{15}x\)) by 336.

We can write this relationship as an equation:

Incorrect calculation − Correct calculation = Difference

\(\frac{2}{5}x - \frac{2}{15}x = 336\)

Solving the Equation for the Unknown Number

To solve the equation \(\frac{2}{5}x - \frac{2}{15}x = 336\), we first need to find a common denominator for the fractions \(\frac{2}{5}\) and \(\frac{2}{15}\). The least common multiple of 5 and 15 is 15.

Rewrite the fractions with the common denominator:

\(\frac{2}{5}x = \frac{2 \times 3}{5 \times 3}x = \frac{6}{15}x\)

Now substitute this back into the equation:

\(\frac{6}{15}x - \frac{2}{15}x = 336\)

Combine the terms on the left side:

\(\left(\frac{6}{15} - \frac{2}{15}\right)x = 336\)

\(\frac{6 - 2}{15}x = 336\)

\(\frac{4}{15}x = 336\)

To find \(x\), we need to isolate it. Multiply both sides of the equation by the reciprocal of \(\frac{4}{15}\), which is \(\frac{15}{4}\):

\(x = 336 \times \frac{15}{4}\)

Now, perform the calculation. We can simplify by dividing 336 by 4:

\(336 \div 4 = 84\)

So, the equation becomes:

\(x = 84 \times 15\)

Calculate the final product:

\(84 \times 15\)

We can do this multiplication:

So, \(x = 1260\).

Verifying the Answer

Let's check if the number 1260 satisfies the original condition.

• Correct calculation: \(\frac{2}{15} \times 1260 = 2 \times \frac{1260}{15} = 2 \times 84 = 168\)
• Incorrect calculation: \(\frac{2}{5} \times 1260 = 2 \times \frac{1260}{5} = 2 \times 252 = 504\)

The difference between the incorrect and correct calculations is \(504 - 168\).
\(504 - 168 = 336\)
This matches the difference given in the problem. Therefore, the number is indeed 1260.

Comparing with Options

The number we found is 1260, which corresponds to one of the given options.

• Option 1: 1680
• Option 2: 1344
• Option 3: 1008
• Option 4: 1260

Our calculated number matches Option 4.

Revision Table: Fractions and Equations

Additional Information: Working with Fractions and Differences

When a problem states that one quantity is "greater than" another by a certain amount, it means the difference between the larger quantity and the smaller quantity is that amount. In this case, the incorrect answer was larger than the correct answer.

The fractions \(\frac{2}{5}\) and \(\frac{2}{15}\) represent parts of the same number. Since \(\frac{2}{5}\) is a larger fraction than \(\frac{2}{15}\) (because 5 is smaller than 15, or by finding a common denominator, \(\frac{6}{15}\) vs \(\frac{2}{15}\)), calculating \(\frac{2}{5}\) of a number will always yield a larger result than calculating \(\frac{2}{15}\) of the same number (for positive numbers).

Setting up the correct equation based on the problem's wording is the crucial first step. The phrase "greater than... by" directly translates to a subtraction leading to the given difference.