Question

A cake chain manufactures two types of products: cakes/pastries/gateaux, and savouries. The chain was concerned about high wastage (in terms of leftover) and wanted to reduce it. The table below gives sales, costs and leftover (as a percentage of sales) for both product lines from 1993 to 2004.

If profit = sales - cost - leftover, in which of these years was the cake chain in losses (leftover here means the leftover amount, not the percentage)?

1. 1993
2. 1997
3. 1998
4. 2000

Choose the right option:

• A. 1, 2, 3, 4
• B. 3, 4
• C. 2, 3
• D. 1, 2, 3

Hint:

Convert each year's leftover percentage into an actual amount, then check whether sales cover cost plus that leftover.

Answer 1

The Correct Option is A

A quicker way to check a loss year is to add up cost and the leftover amount first, call this the real outgo, and see if it beats sales. If real outgo is bigger than sales, the year is a loss.

Take 2000 first. Cakes real outgo = $743.20 + 1.50\% \times 752.09 = 743.20 + 11.28 = 754.48$, against sales of $752.09$, so cakes alone lose $2.39$. Savouries real outgo = $630.09 + 5.61\% \times 637.63 = 630.09 + 35.78 = 665.87$, against sales of $637.63$, a loss of $28.24$. Added together the chain loses $30.63$ in 2000.

Now 1998. Cakes real outgo = $926.06 + 1.74\% \times 936.52 = 926.06 + 16.30 = 942.36$ against sales $936.52$, a loss of $5.84$. Savouries real outgo = $464.60 + 11.04\% \times 466.19 = 464.60 + 51.49 = 516.09$ against sales $466.19$, a loss of $49.90$. Total loss for 1998 is $55.74$.

Now 1997. Cakes real outgo = $865.69 + 1.06\% \times 886.71 = 865.69 + 9.40 = 875.09$ against sales $886.71$, so cakes actually earn $11.62$. Savouries real outgo = $511.10 + 8.29\% \times 516.32 = 511.10 + 42.80 = 553.90$ against sales $516.32$, a loss of $37.58$. Net for 1997 is a loss of $25.96$, because the savouries loss swallows the cakes profit.

Finally 1993. Cakes real outgo = $80.06 + 1.52\% \times 81.47 = 80.06 + 1.24 = 81.30$ against sales $81.47$, a tiny profit of $0.17$. Savouries real outgo = $41.07 + 9.38\% \times 41.79 = 41.07 + 3.92 = 44.99$ against sales $41.79$, a loss of $3.20$. Net for 1993 is a loss of $3.03$.

So every year in the list, 2000, 1998, 1997 and 1993, shows real outgo higher than sales once the leftover is added in. The chain was in loss in all four years listed, statements 1, 2, 3 and 4.
\[ \boxed{1,\,2,\,3,\,4} \]