Chapter 11: Problem 14
If \(A: B\) is \(3: 7, C: D\) is \(15: 11,\) and \(B: C\) is \(14: 5,\) what is \(A: D ?\)
Short Answer
Expert verified
The ratio A:D is 18:11.
Step by step solution
01
- Understand Given Ratios
Identify the given ratios: 1. The ratio of A to B is 3:7 2. The ratio of C to D is 15:11 3. The ratio of B to C is 14:5
02
- Express Ratios as Fractions
Convert the given ratios to fractions: 1. \(\frac{A}{B} = \frac{3}{7}\) 2. \(\frac{C}{D} = \frac{15}{11}\) 3. \(\frac{B}{C} = \frac{14}{5}\)
03
- Find Relationship Between Variables
Using \(\frac{B}{C} = \frac{14}{5}\), rearrange to get C in terms of B:\(C = \frac{5B}{14}\)
04
- Substitute C into C:D Ratio
Substitute \(\frac{5B}{14}\) for C in the ratio \(\frac{C}{D} = \frac{15}{11}\):\(\frac{\frac{5B}{14}}{D} = \frac{15}{11}\)
05
- Solve for D in Terms of B
Cross-multiply and solve for D:\(5B \times 11 = 15 \times 14D\) \(55B = 210D\) \(D = \frac{55B}{210}\) Simplify this to \(D = \frac{11B}{42}\)
06
- Substitute A and D into A:D Ratio
Knowing \(\frac{A}{B} = \frac{3}{7}\) and \(\frac{D}{B} = \frac{11}{42}\), write \(A:D\) as \(A = \frac{3B}{7}\) and \(D = \frac{11B}{42}\)
07
- Express A:D as a Ratio
Combine the two equations to form the ratio \(A:D\):\(A:D = \frac{3B/7}{11B/42} = \frac{3B \times 42}{7 \times 11B}\) Cancel \(B\) and simplify fractions:\(A:D = \frac{3 \times 42}{7 \times 11} = \frac{126}{77} = \frac{18}{11}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
ratio comparisons
When handling ratios, it's important to compare them correctly. A ratio compares how much of one thing there is compared to another. For example, in our problem, the ratio of A to B is 3:7. This means for every 3 units of A, there are 7 units of B.
Comparing ratios directly may not always be straightforward. You might need to relate different ratios to each other. In the original exercise, the task was to relate A : B, C : D, and B : C to find the ratio A : D. One way to handle this is by expressing different ratios with a common variable, which makes it easier to compare and solve.
By comparing and analyzing ratios meticulously, you can establish relationships between variables and solve for unknown values.
Comparing ratios directly may not always be straightforward. You might need to relate different ratios to each other. In the original exercise, the task was to relate A : B, C : D, and B : C to find the ratio A : D. One way to handle this is by expressing different ratios with a common variable, which makes it easier to compare and solve.
By comparing and analyzing ratios meticulously, you can establish relationships between variables and solve for unknown values.
fraction conversion
Understanding how to convert ratios into fractions is crucial for solving ratio problems. A ratio of 3:7 can be expressed as the fraction \(\frac{3}{7}\).
In our exercise, each given ratio—like A : B, C : D, and B : C—was first converted into its fractional form to facilitate actual calculations. This conversion made it easier to manipulate and solve for other variables.
Here's the conversion for all given ratios:
In our exercise, each given ratio—like A : B, C : D, and B : C—was first converted into its fractional form to facilitate actual calculations. This conversion made it easier to manipulate and solve for other variables.
Here's the conversion for all given ratios:
- A : B = 3 : 7 converts to \(\frac{A}{B} = \frac{3}{7}\)
- C : D = 15 : 11 converts to \(\frac{C}{D} = \frac{15}{11}\)
- B : C = 14 : 5 converts to \(\frac{B}{C} = \frac{14}{5}\)
cross-multiplication
Cross-multiplication is a valuable technique to solve ratios expressed as fractions. Let's break down how it works.
When you have \(\frac{A}{B} = \frac{C}{D}\), cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second, and vice versa. This gives you: \A \times D = B \times C\.
In our solution:
When we had \(\frac{B}{C} = \frac{14}{5}\), it was rearranged to solve for C in terms of B:
\(C = \frac{5B}{14}\)
Later, we needed to substitute \(C = \frac{5B}{14}\) into \(\frac{C}{D} = \frac{15}{11}\) and use cross-multiplication:
\(5B \times 11 = 15 \times 14D\)
By multiplying across the terms (cross-multiplying), this step helped to isolate the variable D in terms of B, simplifying the process of finding the final ratio.
When you have \(\frac{A}{B} = \frac{C}{D}\), cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second, and vice versa. This gives you: \A \times D = B \times C\.
In our solution:
When we had \(\frac{B}{C} = \frac{14}{5}\), it was rearranged to solve for C in terms of B:
\(C = \frac{5B}{14}\)
Later, we needed to substitute \(C = \frac{5B}{14}\) into \(\frac{C}{D} = \frac{15}{11}\) and use cross-multiplication:
\(5B \times 11 = 15 \times 14D\)
By multiplying across the terms (cross-multiplying), this step helped to isolate the variable D in terms of B, simplifying the process of finding the final ratio.
simplifying ratios
Simplifying ratios is key to finding the most reduced form of your answer. Once you've performed all necessary calculations, you may end up with a complex fraction. Simplifying it will give you a clear and concise answer.
In the exercise:
After we expressed A and D in terms of B: \( \frac{A}{B} = \frac{3}{7}\) and \( \frac{D}{B} = \frac{11}{42}\), we combined them to \( \frac{3B/7}{11B/42}\).
Cancel common factors (here, B):
\( \frac{3B \times 42}{7 \times 11B} = \frac{126}{77} \).
Finally, this ratio was simplified to \( \frac{18}{11} \), providing a clean and understandable result.
Simplification can involve dividing by the greatest common divisor of the numerator and denominator or canceling out common terms. This process helps to present the ratio in its simplest form, making it easy to interpret.
In the exercise:
After we expressed A and D in terms of B: \( \frac{A}{B} = \frac{3}{7}\) and \( \frac{D}{B} = \frac{11}{42}\), we combined them to \( \frac{3B/7}{11B/42}\).
Cancel common factors (here, B):
\( \frac{3B \times 42}{7 \times 11B} = \frac{126}{77} \).
Finally, this ratio was simplified to \( \frac{18}{11} \), providing a clean and understandable result.
Simplification can involve dividing by the greatest common divisor of the numerator and denominator or canceling out common terms. This process helps to present the ratio in its simplest form, making it easy to interpret.
