If a:b=2:3, b:c=2:4 , c:d=2:5, d:e=5:7 , e:f=7:9 then find a:f

To find the ratio \(a:f\) given the series of ratios between consecutive variables, we first need to express all variables in terms of a common variable. This will allow us to find the overall ratio between \(a\) and \(f\).

Given ratios are:
- \(a:b = 2:3\)
- \(b:c = 2:4 = 1:2\)
- \(c:d = 2:5\)
- \(d:e = 5:7\)
- \(e:f = 7:9\)

### Step-by-Step Solution

1. **Express each ratio in terms of a common variable:**

   Since the ratios are consecutive, we can start by expressing each term relative to \(a\).

   - From \(a:b = 2:3\), let \(a = 2x\) and \(b = 3x\).
   - From \(b:c = 1:2\), with \(b = 3x\), we have \(c = 2 \times \frac{b}{2} = 2 \times 3x = 6x\).
   - From \(c:d = 2:5\), with \(c = 6x\), we have \(d = \frac{5}{2} \times 6x = 15x\).
   - From \(d:e = 5:7\), with \(d = 15x\), we have \(e = \frac{7}{5} \times 15x = 21x\).
   - From \(e:f = 7:9\), with \(e = 21x\), we have \(f = \frac{9}{7} \times 21x = 27x\).

2. **Determine the ratio \(a:f\):**

   Now that all variables are expressed in terms of \(x\), we find:
   \[
   a = 2x, \quad f = 27x
   \]
   Therefore, the ratio \(a:f\) is:
   \[
   a:f = \frac{2x}{27x} = \frac{2}{27}
   \]

Thus, the ratio \(a:f = 2:27\).

### Summary

The ratio \(a:f\) is \(2:27\).

Would you like more details on any of the steps, or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the overall ratio between \(a:e\)?
2. How would you find \(b:f\) given the same set of ratios?
3. Can you determine the ratio \(c:e\) directly from the ratios given?
4. How does expressing all terms in a common variable help in finding overall ratios?
5. If another ratio \(f:g = 3:5\) was added, how would you find \(a:g\)?

**Tip:** When dealing with ratios, it's often helpful to express each variable in terms of a common base variable to simplify finding overall relationships.

Learn how to find the ratio a:f in a series of proportional relationships involving variables a, b, c, d, e, and f. This problem covers ratios and proportions, including step-by-step calculations to determine the ratio a:f. Suitable for students in Grades 7-9.

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