Divide `150` into three parts such that the second number is five-sixths the first and the third number is four-fifths the second. Find the largest part.

Options

A. 60
B. 50
C. 40
D. 55

Answer

To solve the problem of dividing 150 into three parts such that the second number is five-sixths of the first and the third number is four-fifths of the second, we can follow these steps:

### Step 1: Define the Variables
Let the first part be \( x \).

### Step 2: Express the Second Part
According to the problem, the second part is five-sixths of the first part. Therefore, we can express the second part as:
\[ \text{Second part} = \frac{5}{6}x \]

### Step 3: Express the Third Part
The third part is four-fifths of the second part. Substituting the expression for the second part, we have:
\[ \text{Third part} = \frac{4}{5} \left(\frac{5}{6}x\right) = \frac{4 \times 5}{5 \times 6}x = \frac{4}{6}x = \frac{2}{3}x \]

### Step 4: Set Up the Equation
Now, we know that the sum of all three parts must equal 150:
\[ x + \frac{5}{6}x + \frac{2}{3}x = 150 \]

### Step 5: Find a Common Denominator
To combine the terms on the left side, we need a common denominator. The least common multiple of 6 and 3 is 6. Thus, we rewrite \(\frac{2}{3}x\) as \(\frac{4}{6}x\):
\[ x + \frac{5}{6}x + \frac{4}{6}x = 150 \]

### Step 6: Combine the Terms
Now, we can combine the terms:
\[ x + \frac{5}{6}x + \frac{4}{6}x = x + \frac{9}{6}x = \frac{6}{6}x + \frac{9}{6}x = \frac{15}{6}x \]
Thus, we have:
\[ \frac{15}{6}x = 150 \]

### Step 7: Solve for \( x \)
To isolate \( x \), we multiply both sides by \(\frac{6}{15}\):
\[ x = 150 \times \frac{6}{15} \]
Calculating this gives:
\[ x = 150 \times 0.4 = 60 \]

### Step 8: Find the Second and Third Parts
Now that we have \( x \):
- The first part (A) is \( 60 \).
- The second part (B) is:
\[ B = \frac{5}{6} \times 60 = 50 \]
- The third part (C) is:
\[ C = \frac{4}{5} \times 50 = 40 \]

### Step 9: Identify the Largest Part
The three parts are:
- First part (A) = 60
- Second part (B) = 50
- Third part (C) = 40

The largest part is:
\[ \text{Largest part} = 60 \]

### Final Answer
The largest part is **60**.

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