2 Simple Approaches to Excess & Shortage Word Problems

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Excess and Shortage word problems typically present two conditions which illustrate either of the following three scenarios:

1.Both conditions lead to an excess of items

2.Both conditions lead to a shortage of items

3.One condition leads to an excess of items while the other leads to a shortage of items.

While the algebraic approach to solving these kinds of questions may appeal to the more advanced students, it is not the recommended method since most students would not have been introduced to algebra before Primary 6.

Below are two non-algebraic methods for solving different types of excess and shortage word problems.

Method 1: Listing

Example:
The number of marbles Amanda has is between 20 and 60. If she packs them into bags of 6 marbles, she will have 5 marbles left unpacked. If she packs them into bags of 8 marbles, she will need another 3 marbles. What is the smallest possible number of marbles that she has?

This method is used when questions include any of the following:
- A range is given (between 20 and 60)
- A minimum or maximum is asked (smallest possible number)

Solution:
‘If she packs them into bags of 6 marbles, she will have 5 marbles left unpacked.’

This condition depicts an ‘excess’ scenario where there will be some leftovers after packing the marbles into bags of 6.
 
Since the number of bags used for packing is not known, we can list down the multiples of 6 that is near the given range to determine the potential number of packed marbles.

For each multiple, we then add the 5 unpacked marbles to determine the possible total number of marbles.

‘If she packs them into bags of 8 marbles, she will need another 3 marbles.’

This condition depicts a ‘shortage’ scenario where there will not be enough marbles
to fill the last bag.

In the same manner as above, we list down the multiples of 8 that is near the given
range, then subtract the 3 marbles that is short from each multiple to determine the
plausible total marbles.

Based on the two lists of potential total marbles we have obtained, we pick the common values that appear in both lists (29 and 53) and choose the value that best fits the requirement of the question (smallest possible number).

Answer: The smallest possible number of marbles that she has is 29.

Method 2: Model

This method is used when there is a fixed number of items given in the question (i.e. no range is given and no maximum or minimum is asked for).

Example 1 (Excess-Shortage):
Jack has some sweets to distribute to his friends. If he gives each friend 8 sweets, he will need another 32 sweets. If he gives each friend 5 sweets, he will have 7 sweets left. How many sweets does Jack have?

Solution:
Let 1 unit represent the number of friends Jack has.
Case 1: ‘If he gives each friend 8 sweets, he will need another 32 sweets.’
Case 2: ‘If he gives each friend 5 sweets, he will have 7 sweets left.’

In the model above, the actual total number of sweets he possesses is the same in both cases, and is indicated by the thick vertical line.

Case 1:
Since the number of friends he has is 1 unit, to give each of them 8 sweets, the total number of sweets required is 8 units. With a shortage of 32 sweets for this to happen, the length of the bar used to represent 8 units of sweets will exceed the actual total number of sweets he possesses.

Case 2:
With the same 1 unit of friends that he has, and with each of them receiving 5 sweets, the total sweets he will distribute is only 5 units. This leaves him with a balance of 7 sweets to make up for the total sweets he actually has.
From the model, we can conclude that the difference of 3 units from both cases is equal to the sum of 7 and 32.

Answer: Jack has 72 sweets.

Example 2 (Excess-Excess):

Fiona has some money to buy some books. The cost of the books is the same. If she buys 4 books, she will have $2 left. If she buys 3 books instead, she will have $20 left. How much money does Fiona have?

Solution:
Let 1 unit represent the cost of each book.

Case 1: ‘If she buys 4 books, she will have $2 left.’

Case 2: ‘If she buys 3 books instead, she will have $20 left.’

Since both cases depict an excess scenario, the model to illustrate the question will be drawn as shown below.

Answer: Fiona has $74.

Example 3 (Shortage-Shortage):
 
Some stickers are to be shared among a group of girls. If each girl receives 15 stickers, there will be a shortage of 55 stickers. If each girls gets 9 stickers instead, the last girl will only receive 8 stickers. How many stickers are there in total?

Solution:
 
Let 1 unit represent number of girls in the group.

Case 1: ‘If each girl receives 15 stickers, there will be a shortage of 55 stickers’

Case 2: ‘. If each girls gets 9 stickers instead, the last girl will only receive 8 stickers.’

Here, both cases present a shortage scenario. It's important to note that in Case 2, the shortage of stickers is 1 rather than 8. This adjustment arises because each girl is supposed to receive 9 stickers, but the last girl only received 8, thus requiring only 1 additional sticker.

Answer: There are 80 stickers in total.

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