What is the value of \(\rm \frac{X}{Y}\) if \(\rm \frac{X-5Y}{X+5Y}=\frac{7}{13}\).
Finding the Value of X/Y from a Given Equation
The problem asks us to find the value of the ratio \(\frac{X}{Y}\) given the equation involving \(X\) and \(Y\).
The given equation is:
\(\frac{X-5Y}{X+5Y}=\frac{7}{13}\)
To find the value of \(\frac{X}{Y}\), we need to rearrange this equation to isolate the ratio. We can do this by cross-multiplying.
Step-by-Step Solution to Find X/Y
Let's solve the given equation:
- Start with the given equation: \(\frac{X-5Y}{X+5Y}=\frac{7}{13}\)
- Cross-multiply the terms: \(13 \times (X-5Y) = 7 \times (X+5Y)\)
- Distribute the numbers on both sides: \(13X - 13 \times 5Y = 7X + 7 \times 5Y\)
- Simplify the multiplication: \(13X - 65Y = 7X + 35Y\)
- Now, collect the terms involving \(X\) on one side of the equation and the terms involving \(Y\) on the other side. Let's move \(7X\) to the left side and \(-65Y\) to the right side: \(13X - 7X = 35Y + 65Y\)
- Combine the like terms on both sides: \(6X = 100Y\)
- The goal is to find \(\frac{X}{Y}\). To get this ratio, we can divide both sides of the equation by \(Y\) (assuming \(Y \neq 0\)) and by 6. First, divide by \(Y\): \(\frac{6X}{Y} = 100\). Then, divide by 6: \(\frac{X}{Y} = \frac{100}{6}\)
- Finally, simplify the fraction \(\frac{100}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{100 \div 2}{6 \div 2} = \frac{50}{3}\)
Thus, the value of \(\frac{X}{Y}\) is \(\frac{50}{3}\).
Conclusion on the Value of X/Y
By solving the given algebraic equation \(\frac{X-5Y}{X+5Y}=\frac{7}{13}\) through cross-multiplication and rearrangement, we found that the ratio \(\frac{X}{Y}\) is equal to \(\frac{50}{3}\).
This matches one of the provided options.
Revision Table: Key Steps for Solving Ratio Equations
| Step | Description | Application in this Problem |
|---|---|---|
| 1 | Identify the equation involving ratios. | \(\frac{X-5Y}{X+5Y}=\frac{7}{13}\) |
| 2 | Cross-multiply to remove denominators. | \(13(X-5Y) = 7(X+5Y)\) |
| 3 | Expand both sides. | \(13X - 65Y = 7X + 35Y\) |
| 4 | Collect terms with the same variable. | \(13X - 7X = 35Y + 65Y\) |
| 5 | Simplify the equation. | \(6X = 100Y\) |
| 6 | Rearrange to isolate the desired ratio (X/Y). | \(\frac{X}{Y} = \frac{100}{6}\) |
| 7 | Simplify the final ratio. | \(\frac{50}{3}\) |
Additional Information: Working with Ratios and Equations
A ratio is a comparison of two quantities by division. In this problem, we are working with the ratio of two variables, \(X\) and \(Y\).
When solving equations involving ratios or fractions, cross-multiplication is a fundamental technique. It helps convert the fractional equation into a linear equation, which is generally easier to solve.
The principle behind cross-multiplication is that if \(\frac{a}{b} = \frac{c}{d}\) (where \(b \neq 0\) and \(d \neq 0\)), then \(ad = bc\). This is derived by multiplying both sides of the equation by \(bd\).
Once the equation is linear (without fractions), the next step is usually to gather terms containing the variables you are interested in on one side and constant terms or other variables on the other side. This involves using inverse operations (addition/subtraction, multiplication/division).
Simplifying fractions at the end is crucial to present the answer in its simplest form.
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If three-fifths of a number is 54, what is two-ninth of it?
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In his career, a tennis player won 5 matches lost 12 matches and had 3 matches as draw. The fraction of the match he lost in his career is:
Simplify:
\(62\div 5 - \left(\frac{8}{9}\times \frac{18}{7}\right)\times \frac{7}{5}+\frac{5}{4}\times \frac{16}{25}\)
