Question

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

Short answer

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

Step-by-step solution

Write down the given equation

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

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: \[\frac{22}{25} = \frac{7}{5}b - b\]

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: \[\frac{22}{25} = \frac{7}{5}b - \frac{5}{5}b\]

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: \[\frac{22}{25} = \frac{2}{5}b\]

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 \(b\): \[b = \frac{22\cdot5}{25\cdot2}\]

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 \(\frac{11}{5}\).

Key concepts

Equations

Equations are mathematical statements that show the equality between two expressions. They often involve variables, which are symbols that stand in for unknown values. Understanding equations is essential because they allow us to solve for these unknown values by performing operations that maintain equality.
A key aspect of solving equations is performing the same operation on both sides. This ensures the balance, or equality, is maintained. For example, if we add or subtract a number from one side, we must do the same to the other. Similarly, when we multiply or divide, consistency between both sides of the equation is crucial.
In our problem, the equation is: \[b + \frac{22}{25} = \frac{7}{5}b\]The goal is to find the value of the variable \(b\) that makes the equation true. First, we isolate terms involving \(b\), then simplify and solve.
Successfully manipulating and solving equations also involves understanding other mathematical concepts like fractions, which we'll delve into next.

Fractions

Fractions represent parts of a whole and are composed of a numerator and a denominator. In algebra, fractions often occur when dealing with ratios, proportions, and division. They are essential for expressing values that are not whole numbers.
In the equation we are working with, fractions appear on both sides:\[b + \frac{22}{25} = \frac{7}{5}b\]Handling fractions within equations can be tricky, but with the right approach, it becomes manageable. One strategy is to find a common denominator to combine fractions effectively. This involves calculating the least common multiple (LCM) of the denominators involved.

• For example, the fractions \(\frac{7}{5}\) and \(\frac{5}{5}\) share 5 as their LCM, allowing us to simplify further.
• Subtracting fractions means adjusting numerators while keeping a common denominator, as done when combining \(\frac{7}{5}b\) and \(-b\).

Understanding how to work with fractions is vital for equation solving, especially when moving terms and doing arithmetic operations to isolate variables.

Variable Isolation

Variable isolation is the process of rearranging an equation so that a single variable is on one side, and all other terms are on the other. This is the final step in solving equations, where the isolated variable's value becomes apparent.
The concept involves working systematically and logically to "undo" all mathematical operations performed on the variable. In our specific case, the target was to isolate \(b\). This was achieved by:

• First, gathering all terms involving \(b\) on one side and the constant fraction \(\frac{22}{25}\) on the other.
• Subtracting \(b\) from both sides resulted in isolating the fractions involving \(b\).

This resulted in the simplified equation:\[\frac{22}{25} = \frac{2}{5}b\]Variable isolation may also involve multiplying by the reciprocal of coefficients, as seen when we multiplied both sides by \(\frac{5}{2}\) to remove the fraction from \(b\). The ultimate goal is to simplify all terms to reveal the value of the variable, leading to our answer, \(b = \frac{11}{5}\). By mastering variable isolation, solving equations becomes a straightforward and logical process.