Devide 170 into three parts such that the first part is 10 more than the second and its ratio with the third part iis 2: 5

Answer

To solve the problem of dividing 170 into three parts such that the first part is 10 more than the second part and the ratio of the first part to the third part is 2:5, we can follow these steps:

### Step 1: Define the Parts
Let:
- The first part be \( A \)
- The second part be \( B \)
- The third part be \( C \)

### Step 2: Set Up the Relationships
According to the problem:
1. The first part \( A \) is 10 more than the second part \( B \):
\[ A = B + 10 \]
2. The ratio of the first part \( A \) to the third part \( C \) is 2:5:
\[ \frac{A}{C} = \frac{2}{5} \]
This can be expressed as:
\[ A = \frac{2}{5}C \]

### Step 3: Express \( B \) and \( C \) in Terms of \( A \)
From the ratio \( A = \frac{2}{5}C \), we can express \( C \) in terms of \( A \):
\[ C = \frac{5}{2}A \]

Now substituting \( A = B + 10 \) into the equation for \( C \):
\[ C = \frac{5}{2}(B + 10) \]

### Step 4: Write the Equation for the Total
The total of the three parts is given as:
\[ A + B + C = 170 \]

Substituting \( A \) and \( C \) in terms of \( B \):
\[ (B + 10) + B + \frac{5}{2}(B + 10) = 170 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ 2B + 10 + \frac{5}{2}B + 25 = 170 \]
\[ 2B + \frac{5}{2}B + 35 = 170 \]

To eliminate the fraction, multiply the entire equation by 2:
\[ 4B + 5B + 70 = 340 \]
\[ 9B + 70 = 340 \]

### Step 6: Solve for \( B \)
Subtract 70 from both sides:
\[ 9B = 270 \]
Now divide by 9:
\[ B = 30 \]

### Step 7: Find \( A \) and \( C \)
Now that we have \( B \), we can find \( A \) and \( C \):
1. Calculate \( A \):
\[ A = B + 10 = 30 + 10 = 40 \]
2. Calculate \( C \):
\[ C = \frac{5}{2}A = \frac{5}{2} \times 40 = 100 \]

### Step 8: Verify the Total
Now we check if the total adds up to 170:
\[ A + B + C = 40 + 30 + 100 = 170 \]

### Final Answer
Thus, the three parts are:
- First part \( A = 40 \)
- Second part \( B = 30 \)
- Third part \( C = 100 \)

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