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Chapter 2: Problem 34

If \(b+\frac{22}{25}=\frac{7}{5} b\), what is the value of \(b\) ?

Short Answer

Expert verified
The value of \(b\) is \(\frac{11}{5}\).

Step by step solution

01

Write down the given equation

We are given the equation: \(b + \frac{22}{25} = \frac{7}{5}b\).
02

Subtract b from both sides

To isolate the term with the variable \(b\) on one side of the equation, we will subtract \(b\) from both sides. This gives us the equation: \[\frac{22}{25} = \frac{7}{5}b - b\]
03

Find a common denominator for the fractions

To combine the terms on the right side, we need to find a common denominator for the fractions involved. The common denominator for 5 and 1 (since \(b\) can be considered as a fraction with a denominator of 1) can be taken as their lowest common multiple. The LCM of 5 and 1 is 5. Therefore, we can rewrite the equation as: \[\frac{22}{25} = \frac{7}{5}b - \frac{5}{5}b\]
04

Combine the terms on the right side

Now, we can subtract the fractions on the right side by subtracting their numerators and keeping the denominator the same. This gives us: \[\frac{22}{25} = \frac{7-5}{5}b\] Simplifying the right side, we get: \[\frac{22}{25} = \frac{2}{5}b\]
05

Solve for b

To find the value of \(b\), we can multiply both sides of the equation by the reciprocal of the coefficient of \(b\). The reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\), so we will multiply both sides by \(\frac{5}{2}\) and simplify to get: \[\frac{5}{2}\cdot\frac{22}{25} = \frac{5}{2}\cdot\frac{2}{5}b\] On the right side, the factors of 5 and 2 in the numerator and denominator cancel out, leaving us with just \(b\): \[b = \frac{22\cdot5}{25\cdot2}\]
06

Simplify the expression for b

Simplifying the expression for \(b\), we can cancel out the common factors between the numerator and the denominator: \[b = \frac{2\cdot11\cdot5}{5\cdot5\cdot2}\] The common factors of 5 and 2 in the numerator and the denominator cancel out, leaving us with: \[b = \frac{11}{5}\] So the value of \(b\) is \(\frac{11}{5}\).

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