To determine the ratio \( a : c \) given the ratios \( a : b = 3 : 5 \) and \( b : c = 10 : 11 \), we can follow these steps:
1. Understand the given ratios:
- The ratio \( a : b \) is given as \( 3 : 5 \).
- The ratio \( b : c \) is given as \( 10 : 11 \).
2. Convert these ratios into fractions:
- \( a : b = \frac{a}{b} = \frac{3}{5} \)
- \( b : c = \frac{b}{c} = \frac{10}{11} \)
3. Find the product of the fractions to determine \( a : c \):
[tex]\[
\frac{a}{c} = \frac{a}{b} \times \frac{b}{c} = \frac{3}{5} \times \frac{10}{11}
\][/tex]
4. Multiply the numerators and the denominators:
[tex]\[
\frac{a}{c} = \frac{3 \times 10}{5 \times 11} = \frac{30}{55}
\][/tex]
5. Simplify the fraction \(\frac{30}{55}\) to its simplest form:
- The greatest common divisor (GCD) of 30 and 55 is 5.
- Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{30 \div 5}{55 \div 5} = \frac{6}{11}
\][/tex]
Therefore, the ratio \( a : c \) in its simplest form is:
\(
a : c = 6 : 11
\)
Optional working
Answer: [tex]\( a : c = 6 : 11 \)[/tex]
