Problem Analysis: Finding Total Balls

The problem involves finding the total number of balls given the fractions of red and blue balls and the difference between the number of red and yellow balls.

• Fraction of red balls: $\frac{7}{18}$
• Fraction of blue balls: $\frac{8}{15}$
• The rest are yellow balls.
• Number of red balls = Number of yellow balls + 84

Calculating Fractions of Balls

First, find the combined fraction of red and blue balls.

• To add $\frac{7}{18}$ and $\frac{8}{15}$, find the least common multiple (LCM) of the denominators 18 and 15. LCM(18, 15) = 90.
• Convert the fractions:
  • Red: $\frac{7}{18} = \frac{7 \times 5}{18 \times 5} = \frac{35}{90}$
  • Blue: $\frac{8}{15} = \frac{8 \times 6}{15 \times 6} = \frac{48}{90}$
• Combined fraction (Red + Blue): $\frac{35}{90} + \frac{48}{90} = \frac{83}{90}$

Next, find the fraction of yellow balls.

• Fraction of Yellow balls = 1 - (Fraction of Red + Fraction of Blue)
• Fraction of Yellow balls = $1 - \frac{83}{90} = \frac{7}{90}$

Calculating the Difference Fraction

The problem states there are 84 more red balls than yellow balls. Calculate the difference between the fraction of red balls and yellow balls.

• Fraction difference (Red - Yellow) = $\frac{7}{18} - \frac{7}{90}$
• Use the common denominator 90:
  • Red: $\frac{35}{90}$
  • Yellow: $\frac{7}{90}$
• Difference: $\frac{35}{90} - \frac{7}{90} = \frac{28}{90}$
• Simplify the difference fraction: $\frac{28}{90} = \frac{14}{45}$

Calculating Total Number of Balls

The fraction $\frac{14}{45}$ represents the difference of 84 balls.

• Let the total number of balls be T.
• Equation: $\frac{14}{45} \times T = 84$
• Solve for T:
  • $T = 84 \div \frac{14}{45}$
  • $T = 84 \times \frac{45}{14}$
  • $T = (84 \div 14) \times 45$
  • $T = 6 \times 45$
  • $T = 270$

Therefore, the total number of balls in the drum is 270.