In a bag full of small balls, $\frac{1}{4}$ of these balls are green, $\frac{1}{8}$ are blue, $\frac{1}{12}$ are yellow and the remaining $26$ white. How many balls are blue?

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Q. In a bag full of small balls,

\frac{1}{4}

of these balls are green,

\frac{1}{8}

are blue,

\frac{1}{12}

are yellow and the remaining

26

white. How many balls are blue?

Calculate total number of balls: Calculate the total number of balls in the bag by considering the fractions of green, blue, and yellow balls, and the known quantity of white balls. $\newline$ Let the total number of balls be $T$ . $\newline$ We know that $\frac{1}{4}$ are green, $\frac{1}{8}$ are blue, $\frac{1}{12}$ are yellow, and the remaining $26$ are white. $\newline$ So, $\left(\frac{1}{4}\right)T + \left(\frac{1}{8}\right)T + \left(\frac{1}{12}\right)T + 26 = T$
Find common denominator: Find a common denominator for the fractions to combine them. $\newline$ The common denominator for $4$ , $8$ , and $12$ is $24$ . $\newline$ So, $(\frac{6}{24})T + (\frac{3}{24})T + (\frac{2}{24})T + 26 = T$ $\newline$ Combine the fractions: $(\frac{6}{24} + \frac{3}{24} + \frac{2}{24})T + 26 = T$ $\newline$ $(\frac{11}{24})T + 26 = T$
Isolate term with T: Isolate the term with T on one side to solve for T. $\newline$ $(\frac{11}{24})T + 26 = T$ $\newline$ Subtract $(\frac{11}{24})T$ from both sides to get: $\newline$ $26 = T - (\frac{11}{24})T$ $\newline$ $26 = (\frac{24}{24})T - (\frac{11}{24})T$ $\newline$ $26 = (\frac{13}{24})T$
Solve for T: Solve for T by multiplying both sides by the reciprocal of $(\frac{13}{24})$ . $\newline$ $T = 26 \times (\frac{24}{13})$ $\newline$ $T = 2 \times 24$ $\newline$ $T = 48$ $\newline$ So, there are $48$ balls in total.
Calculate number of blue balls: Now that we know the total number of balls, calculate the number of blue balls. $\newline$ $(1/8)$ of $48$ is the number of blue balls. $\newline$ $(1/8) \times 48 = 6$