Calculate class marks and frequency total

Calculate class marks ($x_i$) as midpoint: 10, 30, 50, 70, 90. Sum of frequencies is $2 + 18 + x + 6 + y = 70$.

$x + y = 44$

Set up the mean equation

Sum of $f_i x_i$ is $(2 \times 10) + (18 \times 30) + (x \times 50) + (6 \times 70) + (y \times 90) = 20 + 540 + 50x + 420 + 90y = 980 + 50x + 90y$. Since mean is 54, $\frac{980 + 50x + 90y}{70} = 54$.

$980 + 50x + 90y = 3780$

Simplify and solve for x and y

Subtract 980 from 3780: $50x + 90y = 2800$. Divide by 10: $5x + 9y = 280$. We have $x + y = 44 \implies x = 44 - y$. Substituting this: $5(44 - y) + 9y = 280$.

$220 - 5y + 9y = 280 \implies 4y = 60 \implies y = 15$

Final calculation

Find $x$ using $x + y = 44$: $x = 44 - 15 = 29$. Now calculate $6x + 9y = 6(29) + 9(15) = 174 + 135 = 309$.

$6(29) + 9(15) = 309$