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A garden-supply company mixes a potting blend from peat, sand, and compost in the ratio 5 : 3 : 2 by weight. To one batch of this blend the company adds 40 kilograms of compost and nothing else, after which compost makes up 1/3 of the batch by weight. What was the weight, in kilograms, of the sand in the batch before the compost was added?
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Correct answer
B
Assign one common unit to the ratio, then set up a single fraction equation that updates both the compost amount and the total.
Write the three parts using a common scale factor k: peat = 5k, sand = 3k, compost = 2k, so the original total is 10k. Adding 40 kilograms of compost changes two things at once: the compost becomes 2k + 40, and the total becomes 10k + 40 (the added weight is part of the new batch). The new compost share is 1/3, so:
(2k + 40) / (10k + 40) = 1/3
Cross-multiply: 3(2k + 40) = 10k + 40, which gives 6k + 120 = 10k + 40, so 80 = 4k and k = 20. The sand was 3k = 3(20) = 60 kilograms. Check: original total 200 kg with 40 kg compost; after adding 40 kg, compost is 80 and total is 240, and 80/240 = 1/3.
Why each wrong choice is tempting: (A) 40 is the original compost weight (2k). It comes from solving for k correctly but reporting the wrong part: the question asks for sand, not the compost the condition describes. (C) 80 is the compost weight after the 40 kilograms are added (2k + 40). It reports the wrong quantity at the wrong moment, focusing on compost rather than the original sand. (D) 90 is the classic single-update error. Putting the +40 only in the numerator, (2k + 40)/10k = 1/3, leaves the denominator unchanged and gives k = 30, so sand looks like 90. The total must grow along with the compost. (E) 100 is the peat weight (5k). It uses the right scale factor but reads off the largest original part instead of the sand. Only (B) 60 applies the correct scale factor to the sand after updating both the top and bottom of the fraction.
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