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Chapter 8: Problem 369

In the following exercises, solve. $$ \frac{5}{a}=\frac{65}{117} $$

Short Answer

Expert verified
a = \(\frac{117}{13}\)

Step by step solution

01

- Cross Multiply

To solve the equation \(\frac{5}{a} = \frac{65}{117}\), start by cross multiplying to eliminate the fractions. This gives: \[5 \times 117 = 65 \times a\]
02

- Perform the Multiplications

Calculate the products from the cross multiplication step. \[5 \times 117 = 585\] and \[65 \times a = 65a\]. So the equation becomes: \[585 = 65a\]
03

- Solve for a

To isolate \(a\), divide both sides of the equation by 65. \[a = \frac{585}{65}\]
04

- Simplify the Fraction

Simplify \(\frac{585}{65}\). To do this, find the greatest common divisor (GCD) of 585 and 65, which is 5. So, \[a = \frac{585 \div 5}{65 \div 5} = \frac{117}{13}\]
05

- Write the Final Answer

Thus, the simplified form of \(a\) is \(\frac{117}{13}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross Multiplication
Cross multiplication is a method used to solve equations that involve fractions.
It helps eliminate the fractions, making the equation easier to deal with.
To cross multiply the equation \(\frac{5}{a} = \frac{65}{117}\), follow these steps:
路 Multiply the numerator of the first fraction (5) by the denominator of the second fraction (117).
路 Multiply the numerator of the second fraction (65) by the denominator of the first fraction (a).
This results in the equation: \[5 \times 117 = 65 \times a\].
This step removes the fractions and gives you a simple multiplication equation to solve.
Solving for a Variable
Once you have cross-multiplied, you get an equation that is easier to work with.
Now, you need to isolate the variable you want to solve for.
In our equation from cross-multiplication \([585 = 65a]\), aim to isolate 'a'.
To do this, divide both sides of the equation by the coefficient of 'a' (which is 65).
So, you perform: \[a = \frac{585}{65}\].
This isolates 'a' on one side, leaving you with a simpler fraction to work with.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form.
This helps make the fraction easier to understand and work with.
In our equation, \[a = \frac{585}{65}\], simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
Here, the GCD of 585 and 65 is 5.
Divide the numerator and the denominator by their GCD to simplify the fraction: \[a = \frac{585 \div 5}{65 \div 5} = \frac{117}{13}\].
This gives you the simplest form of the answer, making 'a' easier to understand.

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