Solving Chain Ratios A to D 

The question provides us with a series of ratios connecting different quantities: A, B, C, and D. We are given the ratios A ∶ B, B ∶ C, and C ∶ D, and we need to find the ratio A ∶ D.

The given ratios are:

• A ∶ B = 5 ∶ 4, which can be written as $\frac{A}{B} = \frac{5}{4}$
• B ∶ C = 6 ∶ 5, which can be written as $\frac{B}{C} = \frac{6}{5}$
• C ∶ D = 7 ∶ 10, which can be written as $\frac{C}{D} = \frac{7}{10}$

To find the ratio A ∶ D, we can use the property of ratios where we can multiply consecutive ratios. When we multiply $\frac{A}{B}$, $\frac{B}{C}$, and $\frac{C}{D}$, the intermediate terms B and C cancel out, leaving us with $\frac{A}{D}$.

So, we have:

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

Now, substitute the given values of the ratios into this equation:

$\frac{A}{D} = \frac{5}{4} \times \frac{6}{5} \times \frac{7}{10}$

We can simplify this expression by canceling out common factors in the numerator and the denominator before multiplying:

• The '5' in the numerator of the first fraction and the denominator of the second fraction cancel out.
• The '6' in the numerator and the '4' in the denominator can both be divided by 2, resulting in 3 and 2 respectively.

Let's perform the cancellations:

$\frac{A}{D} = \frac{\cancel{5}}{4} \times \frac{6}{\cancel{5}} \times \frac{7}{10}$

$\frac{A}{D} = \frac{1}{4} \times \frac{6}{1} \times \frac{7}{10}$

Now simplify further by dividing 6 and 4 by 2:

$\frac{A}{D} = \frac{1}{\cancel{4}_2} \times \frac{\cancel{6}_3}{1} \times \frac{7}{10}$

$\frac{A}{D} = \frac{1}{2} \times \frac{3}{1} \times \frac{7}{10}$

Now, multiply the remaining terms in the numerator and the denominator:

Numerator: $1 \times 3 \times 7 = 21$

Denominator: $2 \times 1 \times 10 = 20$

So, we get:

$\frac{A}{D} = \frac{21}{20}$

This fraction represents the ratio A ∶ D.

Therefore, A ∶ D = 21 ∶ 20.

Let's check the options provided:

• 21 ∶ 25
• 7 ∶ 3
• 21 ∶ 20
• 25 ∶ 24

Our calculated ratio 21 ∶ 20 matches one of the options.

The final answer is 21 ∶ 20.

Revision Table: Summary of Given Ratios

Additional Information on Combining Ratios

When you have a chain of ratios like A:B, B:C, C:D, you can find a combined ratio A:B:C:D or a direct ratio like A:D or B:D. To find A:D, as shown above, you can multiply the fractions A/B, B/C, and C/D. This works because the intermediate terms cancel out:

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

Alternatively, you could find A:B:C first and then combine it with C:D to find A:B:C:D. To find A:B:C from A:B = 5:4 and B:C = 6:5, you need to make the 'B' term common. The least common multiple (LCM) of 4 and 6 is 12.

• A:B = 5:4 = (5 $\times$ 3) : (4 $\times$ 3) = 15:12
• B:C = 6:5 = (6 $\times$ 2) : (5 $\times$ 2) = 12:10

So, A:B:C = 15:12:10.

Now combine A:B:C = 15:12:10 with C:D = 7:10. Here, the common term is 'C'. The values for C are 10 and 7. The LCM of 10 and 7 is 70.

• A:B:C = 15:12:10 = (15 $\times$ 7) : (12 $\times$ 7) : (10 $\times$ 7) = 105:84:70
• C:D = 7:10 = (7 $\times$ 10) : (10 $\times$ 10) = 70:100

So, A:B:C:D = 105:84:70:100.

From A:B:C:D = 105:84:70:100, we can find A:D by taking the first and last terms:

A:D = 105:100

This ratio can be simplified by dividing both numbers by their greatest common divisor, which is 5:

A:D = (105 ÷ 5) : (100 ÷ 5) = 21:20

Both methods yield the same result, A:D = 21:20. The multiplication method is often quicker for finding a direct ratio like A:D from a chain.