Given that the ratio $A : B : C = 6 : 7 : 9$ and that $B = 21$, we can solve for $A$ and $C$.

Step 1: Set up the ratios

Since the ratio $A : B : C = 6 : 7 : 9$, we know that: $\frac{A}{6} = \frac{B}{7} = \frac{C}{9}$

Step 2: Substitute $B = 21$

Using $\frac{B}{7} = 21$, we can find the value of each part in the ratio.

$\frac{B}{7} = 21 \Rightarrow 1 \text{ part} = \frac{21}{7} = 3$

Step 3: Solve for $A$ and $C$

Now that we know each part is 3:

• $A = 6 \times 3 = 18$
• $C = 9 \times 3 = 27$

Solution

So, the values of $A$ and $C$ are: $A = 18 \quad \text{and} \quad C = 27$

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. What if $C$ were given instead of $B$?
2. How do we check if a ratio is proportional when only given partial values?
3. Can we use similar methods to solve for unknowns in 4-part ratios?
4. How do ratios relate to proportions in word problems?
5. How would you scale these values if the ratio sum changed?

Tip: Always simplify ratios first when possible to make calculations easier.