A man sells different brands of an items. ^1/ 9 of the items he has in his shop are from Brand A, ^5/ 8 of the…

1. Analyze the question

We need to find the total number of items in the shop first. We are given:

• Brand A = $1/9$ of the total.
• Brand B = $5/8$ of the remainder.
• Brand C = 81 items.
• Goal: Find (Brand B - Brand C).

2. Explain the concept

Let the total number of items be $x$.

• If Brand A is $1/9$ of $x$, then the remainder is $x - 1/9x = 8/9x$.
• Brand B is $5/8$ of that remainder ($8/9x$).
• Brand C is what is left after taking out Brand A and Brand B.

3. Show the solution

Step 1: Find the number of items for each brand in terms of $x$.

• Brand A: $\frac{1}{9}x$
• Remainder after A: $x - \frac{1}{9}x = \frac{8}{9}x$
• Brand B: $\frac{5}{8} \times (\frac{8}{9}x) = \frac{5}{9}x$
• Brand C: The total minus (Brand A + Brand B): $x - (\frac{1}{9}x + \frac{5}{9}x) = x - \frac{6}{9}x = x - \frac{2}{3}x = \frac{1}{3}x$

Step 2: Solve for $x$ using the given value for Brand C. We know Brand C = 81. $\frac{1}{3}x = 81$ $x = 81 \times 3 = 243$

Step 3: Calculate the number of items for Brand B. Brand B = $\frac{5}{9}x$ Brand B = $\frac{5}{9} \times 243$ $243 \div 9 = 27$ $5 \times 27 = 135$

Step 4: Find the difference between Brand B and Brand C. Difference = Brand B - Brand C Difference = $135 - 81 = 54$

4. Identify the answer

Comparing our result to the options provided: A. 243 B. 108 C. 54 D. 135

The correct option is C.

5. Tip

When dealing with "remainder" problems, always calculate the remainder first before applying the next fraction. A common mistake is to apply the second fraction ($5/8$) to the total instead of the remainder. Also, notice how the fractions ($1/9$ and $5/9$) sum up to $6/9$, leaving $3/9$ (or $1/3$) for the final category—this is a quick way to verify your work!