IMAT 2011 Q30 [Apples]

Apples cost 30c each, bananas cost 40c each and oranges cost 50c each. Daniel spends exactly €2 buying fruit.

Which of the statements below are correct?

  1. He cannot have exactly 3 apples.
  2. He must have at least one banana.
  3. He has 4, 5 or 6 fruits.
  4. If he has all 3 types of fruit, he must have fewer apples than bananas and oranges

A. 1, 2 and 3 only

B. 1, 2 and 4 only

C. 1, 3 and 4 only

D. 2, 3 and 4 only

E. 1, 2, 3 and 4

Some Quick Questions to ask yourself when doing this type of problem solving:

  • What is being asked? Underline the specifics in the question.
  • Analyse the data, is there anything we can eliminate immediately?
  • What information do I need to solve the question? Where can I find it?
  • How can I draw or use the graph to help me?

Here, the best approach is to try and disprove the statements.

  1. Let’s take 3 apples at a cost of €0.30 each = €0.90. We have €1.10 left over from his total €2 spent.

Banana + banana = €0.80, which is not enough. If we add another banana = €1.20 which is too much.

Orange + Orange = €1.00, which is not enough, but adding another orange would be too much.

Orange + Banana = €0.90, which is not enough, if we add another banana or orange it would be too much.

Therefore the original statement for 1. is true.

  1. This is not true and we can easily disprove this by taking:

Orange + orange + orange + orange = €2.00

Therefore this is false.

  1. This can be tested by taking the cheapest fruit and the expensive fruit. Then we can have a good range. At the bottom of the range: Apple + Apple + Apple + Apple + Apple = €1.50. Add one final orange to get to the target of €2.00 and we have 6 fruits.

Orange + orange + orange + orange = €2.00. This has four fruits total.

This is probably enough to be conclusive but just to double check we can find a five fruit combination. Apple + Apple + Banana + Orange + Orange = $€2.00.

Therefore this statement is correct.

  1. Start by taking one of each type: €0.30 + €0.40 + €0.50 = €1.20

We can then add an apple since we are trying to see if we can have more apples than bananas and oranges combined.

€1.20 + €0.30 = €1.50. We have only €0.50 left to spend, therefore we can only get an orange. Any apple or banana would not leave us with a total of €2.00 left. We will have 2 apples, 1 banana, and 2 oranges, which means less apples than oranges and bananas.

Therefore this is true.

So We know that 1, 3, and 4 are correct, which is C.