Answer:
To find the ratio \( a:b:c \), we need to first ensure that the middle term \( b \) is the same in both ratios.
Given:
1. \( a:b = 3:5 \)
2. \( b:c = 6:7 \)
We can see that the middle term \( b \) in the first ratio is the same as the first term in the second ratio.
To align these ratios, we can express both ratios in terms of a common middle term \( b \).
From the first ratio, we have \( a:b = 3:5 \). This implies \( b = \frac{5}{3}a \).
Substitute this expression for \( b \) into the second ratio:
\( b:c = 6:7 \) becomes \( \left(\frac{5}{3}a\right):c = 6:7 \)
Now, we have two ratios both in terms of \( a \) and \( c \).
To make the second term of both ratios the same, we need to adjust the first ratio. Since \( b = \frac{5}{3}a \), we can find \( a \) in terms of \( b \) by rearranging the equation: \( a = \frac{3}{5}b \).
So, our adjusted first ratio becomes \( a:b = \frac{3}{5}:1 \).
Now, we can combine both ratios:
\( a:b:c = \frac{3}{5}:1:\frac{7}{6} \)
To simplify, we can multiply all terms by 30 to eliminate the fractions:
\( a:b:c = 18:30:35 \)
