Finding the Ratio A:D from Given Equations

This problem asks us to find the ratio of A to D, given a series of equations relating A, B, C, and D. We are given the following relationships:

• 5A = 4B
• 7B = 3C
• 2C = 7D

To find the ratio A:D, we can first express each given equation as a ratio and then combine them. Let's convert each equation into a ratio form:

Step 1: Convert Equations to Ratios

From the first equation, 5A = 4B, we can find the ratio A:B:

$$ \frac{A}{B} = \frac{4}{5} $$

So, the ratio A:B is 4:5.

From the second equation, 7B = 3C, we can find the ratio B:C:

$$ \frac{B}{C} = \frac{3}{7} $$

So, the ratio B:C is 3:7.

From the third equation, 2C = 7D, we can find the ratio C:D:

$$ \frac{C}{D} = \frac{7}{2} $$

So, the ratio C:D is 7:2.

Step 2: Chain the Ratios to Find A:D

Now that we have the individual ratios A:B, B:C, and C:D, we can chain them together to find the ratio A:D. We can do this by multiplying the fractional forms of the ratios:

$$ \frac{A}{D} = \left(\frac{A}{B}\right) \times \left(\frac{B}{C}\right) \times \left(\frac{C}{D}\right) $$

Substitute the values of the ratios we found:

$$ \frac{A}{D} = \left(\frac{4}{5}\right) \times \left(\frac{3}{7}\right) \times \left(\frac{7}{2}\right) $$

Step 3: Simplify the Expression

Now, we perform the multiplication and simplify the resulting fraction:

$$ \frac{A}{D} = \frac{4 \times 3 \times 7}{5 \times 7 \times 2} $$

We can cancel out common factors in the numerator and denominator. The number 7 appears in both the numerator and denominator. The number 4 in the numerator can be written as $2 \times 2$, allowing us to cancel out one factor of 2 with the 2 in the denominator.

$$ \frac{A}{D} = \frac{4 \times 3 \times \cancel{7}}{5 \times \cancel{7} \times 2} = \frac{4 \times 3}{5 \times 2} $$

Further simplifying:

$$ \frac{A}{D} = \frac{(2 \times 2) \times 3}{5 \times 2} = \frac{2 \times \cancel{2} \times 3}{5 \times \cancel{2}} = \frac{2 \times 3}{5} $$

$$ \frac{A}{D} = \frac{6}{5} $$

Resulting Ratio A:D

The ratio A:D is therefore 6:5.

This can be summarized in a table showing the steps:

The final ratio of A:D is 6:5.

Revision Table: Ratio Calculation Steps

Additional Information: Understanding Ratios and Proportions

A ratio is a comparison of two quantities. It can be written as A:B or as a fraction A/B. Proportions are equations stating that two ratios are equal.

In this problem, we used the property of chaining ratios. If we have ratios A:B, B:C, C:D, and so on, we can find the ratio A:D by multiplying the corresponding fractions:

$$ \frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D} $$

This works because the intermediate terms (B and C in this case) cancel out:

$$ \frac{A}{\cancel{B}} \times \frac{\cancel{B}}{\cancel{C}} \times \frac{\cancel{C}}{D} = \frac{A}{D} $$

This method is very useful for solving problems involving multiple interconnected ratios.