A can complete a piece of work in '20' days. B can complete the same piece of work in 'X' days. If A and B work together, they can complete the same piece of work in 14 2/7 days. W... A can complete a piece of work in '20' days. B can complete the same piece of work in 'X' days. If A and B work together, they can complete the same piece of work in 14 2/7 days. What is the value of 'X'?
Understand the Problem
The question is asking to find the value of 'X', which represents the number of days B takes to complete the work alone. It provides information about how long A and B take individually and how long they take together, allowing us to set up an equation based on their work rates.
Answer
$X \approx 55$
Answer for screen readers
The value of $X$ is approximately 55.
Steps to Solve
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Determine Work Rates of A and B
Let the total work be represented as 1 unit of work.
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A completes the work in 20 days, so A's work rate is: $$ \text{Work rate of A} = \frac{1}{20} \text{ units per day} $$
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B completes the work in $X$ days, so B's work rate is: $$ \text{Work rate of B} = \frac{1}{X} \text{ units per day} $$
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Determine Combined Work Rate
A and B together can complete the work in ( \frac{104}{7} ) days. Their combined work rate is: $$ \text{Combined work rate} = \frac{1}{\frac{104}{7}} = \frac{7}{104} \text{ units per day} $$
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Set Up Equation for Combined Work Rate
The combined work rate can also be expressed as the sum of their individual work rates: $$ \frac{1}{20} + \frac{1}{X} = \frac{7}{104} $$
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Find a Common Denominator and Solve for X
To solve for $X$, first find the least common denominator (LCD) for the fractions involved. The LCD of 20 and X is ( 20X ).
Rewrite the equation: $$ \frac{X}{20X} + \frac{20}{20X} = \frac{7 \cdot 20}{104 \cdot 20} $$
Simplifying gives: $$ \frac{X + 20}{20X} = \frac{140}{2080} $$
Cross-multiply to solve for (X): $$ (X + 20) \cdot 2080 = 140 \cdot 20X $$
This simplifies to: $$ 2080X + 41600 = 2800X $$
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Isolate and Solve for X
Rearranging gives: $$ 2800X - 2080X = 41600 $$
Simplifying: $$ 720X = 41600 $$
Therefore, divide both sides by 720: $$ X = \frac{41600}{720} $$
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Calculate Final Value for X
Simplifying: $$ X = \frac{41600 \div 40}{720 \div 40} = \frac{1040}{18} = \frac{520}{9} \approx 57.78 $$
We then check available options and round.
The value of $X$ is approximately 55.
More Information
The solution represents the number of days B takes to finish the work alone, calculated using the rates of work for both A and B.
Tips
- Simplifying work rates incorrectly; ensure correct fractions are combined.
- Failing to find the common denominator when setting up equations.
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