Which one of the following fractions will have minimum change in its value if 3 is added to both the numerator and the denominator of all the fractions?

Question

Prepp Team · Accepted Answer

Solution: Minimum Change in Fraction Value

This question asks us to determine which of the given fractions ($\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{6}$) will experience the least change in value when the number 3 is added to both the numerator and the denominator of each fraction.

Analyzing the Effect of Adding to Numerator and Denominator

Let's consider a general fraction represented as $\frac{a}{b}$. When a positive number $k$ is added to both the numerator ($a$) and the denominator ($b$), the new fraction becomes $\frac{a+k}{b+k}$.

The change in the fraction's value is the difference between the new value and the original value:

\[ \text{Change} = \frac{a+k}{b+k} - \frac{a}{b} \]

To calculate this difference, we find a common denominator, which is $b(b+k)$:

\[ \text{Change} = \frac{b(a+k) - a(b+k)}{b(b+k)} \]

Expanding the terms in the numerator gives:

\[ \text{Change} = \frac{ab + bk - ab - ak}{b(b+k)} \]

Simplifying the numerator by canceling out $ab$ terms:

\[ \text{Change} = \frac{bk - ak}{b(b+k)} \]

Factoring out $k$ from the numerator yields the formula for the change:

\[ \text{Change} = \frac{k(b-a)}{b(b+k)} \]

In this specific problem, we are adding $k=3$ to both the numerator and the denominator. So, the formula becomes:

\[ \text{Change} = \frac{3(b-a)}{b(b+3)} \]

Our goal is to find the fraction $\frac{a}{b}$ from the options that results in the minimum value for this expression.

Calculating Change for Each Fraction

We will now apply the change calculation with $k=3$ to each given fraction. Note that for all the given fractions ($\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$), the numerator ($a$) is less than the denominator ($b$). This means $b-a$ is positive. Since $k=3$ and $b$ are also positive, the change $\Delta$ will always be positive, indicating an increase in the fraction's value.

Fraction 1: $\frac{2}{3}$

Identify numerator ($a$) and denominator ($b$): $a=2$, $b=3$.
Original value: $\frac{2}{3}$.
New value after adding 3: $\frac{2+3}{3+3} = \frac{5}{6}$.
Calculate the change: $\Delta = \frac{5}{6} - \frac{2}{3} = \frac{5}{6} - \frac{4}{6} = \frac{1}{6}$.

Fraction 2: $\frac{3}{4}$

Identify numerator ($a$) and denominator ($b$): $a=3$, $b=4$.
Original value: $\frac{3}{4}$.
New value after adding 3: $\frac{3+3}{4+3} = \frac{6}{7}$.
Calculate the change: $\Delta = \frac{6}{7} - \frac{3}{4} = \frac{24}{28} - \frac{21}{28} = \frac{3}{28}$.

Fraction 3: $\frac{4}{5}$

Identify numerator ($a$) and denominator ($b$): $a=4$, $b=5$.
Original value: $\frac{4}{5}$.
New value after adding 3: $\frac{4+3}{5+3} = \frac{7}{8}$.
Calculate the change: $\Delta = \frac{7}{8} - \frac{4}{5} = \frac{35}{40} - \frac{32}{40} = \frac{3}{40}$.

Fraction 4: $\frac{5}{6}$

Identify numerator ($a$) and denominator ($b$): $a=5$, $b=6$.
Original value: $\frac{5}{6}$.
New value after adding 3: $\frac{5+3}{6+3} = \frac{8}{9}$.
Calculate the change: $\Delta = \frac{8}{9} - \frac{5}{6} = \frac{16}{18} - \frac{15}{18} = \frac{1}{18}$.

Comparing the Changes

We have calculated the change in value for each fraction:

For $\frac{2}{3}$: Change = $\frac{1}{6}$
For $\frac{3}{4}$: Change = $\frac{3}{28}$
For $\frac{4}{5}$: Change = $\frac{3}{40}$
For $\frac{5}{6}$: Change = $\frac{1}{18}$

To determine which change is the minimum, we can compare these fractions. Converting them to decimals makes comparison easier:

$\frac{1}{6} \approx 0.1667$
$\frac{3}{28} \approx 0.1071$
$\frac{3}{40} = 0.075$
$\frac{1}{18} \approx 0.0556$

From the decimal values, it is clear that $\frac{1}{18}$ is the smallest value. This corresponds to the change experienced by the fraction $\frac{5}{6}$.

Alternative Perspective: Fraction Magnitude

Consider the formula for the change: $\Delta = \frac{k(b-a)}{b(b+k)}$. Since $k=3$ is a positive constant, the change $\Delta$ will be minimized when the term $\frac{b-a}{b}$ is minimized.

The term $\frac{b-a}{b}$ can be rewritten as $1 - \frac{a}{b}$. Minimizing $1 - \frac{a}{b}$ is equivalent to maximizing the original fraction $\frac{a}{b}$.

Let's compare the values of the original fractions:

$\frac{2}{3} \approx 0.667$
$\frac{3}{4} = 0.750$
$\frac{4}{5} = 0.800$
$\frac{5}{6} \approx 0.833$

The fraction with the largest value is $\frac{5}{6}$. Therefore, adding the same number (3) to both the numerator and the denominator of the largest fraction ($\frac{5}{6}$) results in the smallest change in its value.

Conclusion

By calculating the change for each fraction or by comparing the magnitude of the original fractions, we find that the fraction $\frac{5}{6}$ undergoes the minimum change in value when 3 is added to both its numerator and denominator. The change for $\frac{5}{6}$ is $\frac{1}{18}$, which is the smallest calculated change.

Prepp Team · Answer

$\frac{2}{3}$

Prepp Team · Answer

$\frac{3}{4}$

Prepp Team · Answer

$\frac{4}{5}$

Solution: Minimum Change in Fraction Value

Analyzing the Effect of Adding to Numerator and Denominator

Calculating Change for Each Fraction

Comparing the Changes

Alternative Perspective: Fraction Magnitude

Conclusion

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