`70%` of a number is equal to four fifth of another number. If the difference between the two numbers is 100. What is the biggest number?

Options

A. 700
B. 750
C. 800
D. Cannot be determined

Answer

To solve the problem step by step, let's define the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations
According to the problem:
1. \( 70\% \) of \( x \) is equal to \( \frac{4}{5} \) of \( y \).
2. The difference between the two numbers is \( 100 \).

We can express these statements mathematically:
\[ 0.7x = \frac{4}{5}y \quad \text{(1)} \]
\[ x - y = 100 \quad \text{(2)} \]

### Step 2: Rearrange equation (2)
From equation (2), we can express \( x \) in terms of \( y \):
\[ x = y + 100 \quad \text{(3)} \]

### Step 3: Substitute equation (3) into equation (1)
Now, we will substitute equation (3) into equation (1):
\[ 0.7(y + 100) = \frac{4}{5}y \]

### Step 4: Expand and simplify
Expanding the left-hand side:
\[ 0.7y + 70 = \frac{4}{5}y \]

To eliminate the fraction, we can multiply the entire equation by \( 5 \):
\[ 5(0.7y + 70) = 4y \]
\[ 3.5y + 350 = 4y \]

### Step 5: Rearrange to isolate \( y \)
Now, we will move \( 3.5y \) to the right side:
\[ 350 = 4y - 3.5y \]
\[ 350 = 0.5y \]

### Step 6: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{350}{0.5} = 700 \]

### Step 7: Substitute back to find \( x \)
Now that we have \( y \), we can find \( x \) using equation (3):
\[ x = y + 100 = 700 + 100 = 800 \]

### Conclusion
The biggest number is \( 800 \).