The difference between 4/5 of a number and `45%` of the number is 56. What is `65%` of the number?

Options

A. 96
B. 104
C. 112
D. 120

Answer

To solve the problem, we need to find the number based on the information given in the question. Let's denote the unknown number as \( x \).

### Step 1: Write the expressions for \( \frac{4}{5} \) of the number and \( 45\% \) of the number.

- \( \frac{4}{5} \) of the number \( x \) can be expressed as:
\[ \frac{4}{5} x \]

- \( 45\% \) of the number \( x \) can be expressed as:
\[ 0.45 x \]

### Step 2: Set up the equation based on the given information.

According to the problem, the difference between \( \frac{4}{5} x \) and \( 0.45 x \) is 56. Therefore, we can write the equation as:
\[ \frac{4}{5} x - 0.45 x = 56 \]

### Step 3: Convert \( 0.45 \) to a fraction for easier calculation.

We know that:
\[ 0.45 = \frac{45}{100} = \frac{9}{20} \]

### Step 4: Rewrite the equation using fractions.

Now, we can rewrite the equation:
\[ \frac{4}{5} x - \frac{9}{20} x = 56 \]

### Step 5: Find a common denominator and simplify.

The common denominator between \( 5 \) and \( 20 \) is \( 20 \). We can express \( \frac{4}{5} \) as \( \frac{16}{20} \):
\[ \frac{16}{20} x - \frac{9}{20} x = 56 \]

Now, combine the fractions:
\[ \frac{16 - 9}{20} x = 56 \]
\[ \frac{7}{20} x = 56 \]

### Step 6: Solve for \( x \).

To isolate \( x \), multiply both sides by \( \frac{20}{7} \):
\[ x = 56 \times \frac{20}{7} \]
\[ x = 56 \times \frac{20}{7} = 160 \]

### Step 7: Find \( 65\% \) of the number.

Now that we have the number \( x = 160 \), we can find \( 65\% \) of \( x \):
\[ 65\% \text{ of } x = 0.65 \times 160 \]
\[ = 104 \]

### Final Answer:
Thus, \( 65\% \) of the number is \( 104 \).

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