A wooden beam is cut into three parts whose lengths are in the proportion $\frac{3}{7}:\frac{2}{5}:\frac{3}{4}$. If the entire beam measures 8 m in total, calculate the length of the longest piece.

The correct answer is 3.8 m.

Step 1: Understand the Given Ratio

The lengths of the three parts of the wooden beam are in the proportion:

$\frac{3}{7}:\frac{2}{5}:\frac{3}{4}$

Step 2: Convert the Fractional Ratio to Whole Numbers

To convert this fractional ratio into a simpler whole-number ratio, we find the Least Common Multiple (LCM) of the denominators 7, 5, and 4.

LCM(7, 5, 4) = 140

Now, multiply each fraction in the proportion by 140:

First part: $\frac{3}{7}\times 140=3\times 20=60$

Second part: $\frac{2}{5}\times 140=2\times 28=56$

Third part: $\frac{3}{4}\times 140=3\times 35=105$

So, the ratio of the three parts in simplified whole numbers is 60 : 56 : 105.

Step 3: Calculate the Total Number of Ratio Units

Sum the parts of the whole-number ratio:

Total units = 60 + 56 + 105 = 221

Step 4: Identify the Longest Piece

Comparing the ratio parts (60, 56, and 105), the third part with 105 units is the longest piece.

Step 5: Calculate the Length of the Longest Piece

Given that the entire beam measures 8 m in total, the length of the longest piece is:

$\text{Length}=\frac{105}{221}\times 8\text{m}$

$\text{Length}=\frac{840}{221}\text{m}\approx 3.8009\text{m}$

Rounding to one decimal place gives 3.8 m.