5-Steps to Achieving Mastery in the Assumption Method
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The assumption method (also known as the supposition method) is a faster and more efficient alternative to the guess-and-check method learnt in Primary 3.
In contrast to the guess-and-check method where we make possible predictions within the parameters of the question and check if our predictions are correct, the assumption method adopts a systematic approach that starts off by assuming that the total consists of entirely one item only.
Why Assumption Method is preferred over Guess-and-Check?
The guess-and-check method requires some logical thinking in order to make the next intelligent guess that is closer to the desired answer.
Blindly guessing and checking at every step may be meaningless and time-consuming, especially when numbers in the question are big.
The assumption method, on the other hand, is a fuss-free 5-step method that is guaranteed to give the correct answer without having to spend all the time and efforts into repetitive calculations that we otherwise would have done in the guess-and-check method.
Video solutions:
How does the Assumption Method work?
The 5 steps (ATEDO) for the assumption method are:
1. Assume (A)
We first assume that all the items are made up of only one type.
2. Total (T)
We then find the total based on our assumption. This will inevitably be different from the actual total given in the question.
3. Excess (E)
We then find out how far off we are from the actual total given in the question. With this, we will make replacements between the types of items to make up for this excess.
4. Difference (D) (1-to-1 exchange)
We find out how much of the excess we are accounting for when we replace only one item from our assumption with one of the other type of item.
5. Opposite (O)
Finally, we look for the number of replacements we need to make to account for the excess so as to reach the actual total given in the question. We can do this by taking Excess divided by Difference.
Types of Assumption Method Questions
1. Basic
Example:
There are 10 cats and ducks in a farm. There are 32 legs in total. How many ducks are there?
This type of questions contain:
– 2 types of items (cats and ducks)
– 2 totals (10 animals, 32 legs)
Solution:
1. Assume that all the 10 animals are cats.
We first assume that all the animals are the opposite of what the question is asking for. This is so that we can get to the answer straightaway at the final “opposite” step.
2. Total legs = 10 × 4 = 40
With 10 cats, we will have 40 legs in the farm. Of course, this is not the right answer as the total legs given in the question is 32.
3. Excess = 40 – 32 = 8
We are 8 legs away from the actual total number of legs. As such, not all the animals are cats; there must be some ducks in the farm as well. We will proceed to replace one cat with one duck to see how much closer we are to the correct total number of legs.
4. Difference = 4 – 2 = 2
When we replace one cat with one duck, our total legs drop to 38, which is 2 less than what we obtained from our assumption. Consequently, for every cat that we replace with a duck, the total number of legs will decrease by 2.
5. Opposite (number of ducks) = 8 ÷ 2 = 4
To decrease the total number of legs by 8, with each replacement reducing the total by 2 legs, four replacements are necessary so that there are 32 legs in the farm altogether.
Answer: There are 4 ducks in the farm.
2. ‘Penalty’
Example:
In a spelling bee, 5 points were awarded for each correct spelling and 2 points were deducted for each wrong spelling. Sally spelt 30 words and scored 101 points altogether. How many words did she spell wrongly?
This type of questions contain:
– 2 types of items (words spelt correctly and wrongly)
– 2 totals (30 words, 101 points)
– ‘Penalty’ involved (deduction of points for wrong spelling)
Solution:
1.Assume that all the 30 words were spelt correctly.
We first assume that all the words were of the positive outcome.
2.Total points = 30 × 5 = 150
With 30 words spelt correctly, we would obtain 150 points. This is not the same as the actual total points given which is 101.
3.Excess = 150 – 101 = 49
We are 49 points away from the actual total points. This means that some words must have been spelt wrongly. We will proceed to replace one correctly-spelt word with one wrongly-spelt word to determine how this affects the total points.
4.Difference = 5 + 2 = 7
Misspelling a word would result in a net loss of 7 points — 5 points for the correct spelling foregone and an additional 2 points for the deduction. Consequently, for every replacement made, the total number of points would drop by 7.
5.Opposite (words spelt wrongly) = 49 ÷ 7 = 7
Given that each replacement reduces the total by 7 points, seven replacements are needed to decrease the total points by 49 so that it becomes 101 as per the question.
Answer: Sally spelt 7 words wrongly.
3. Overall Difference Given
Example:
There were 42 children in a class. Each boy was given 4 stickers and each girl was given 3 stickers. The girls received 14 stickers more than the boys. How many boys were there in the class?
This type of questions contain:
– 2 types of items (boys and girls)
– 1 total (42 children)
– Overall difference given (The girls received 14 more stickers than the boys.)
Solution:
1.Assume that all the 42 children were girls.
We first assume that all the children were the one with the overall greater number of stickers.
2.Total stickers = 42 × 3 = 126
42 girls would receive a total of 126 stickers.
3.Excess = 126 – 14 = 112
Since we assumed there to be 42 girls who received a total of 126 stickers, there would be no boys in the class, and their total stickers would be zero. As a result, the difference between the stickers that all the girls received and the stickers that all the boys received would be 126.
Since the actual overall difference given in the question is 14, we need to account for the 112 stickers in excess that we have by replacing some girls with some boys.
4.Difference = 4 + 3 = 7
Each girl-to-boy replacement would mean that the total number of stickers all the girls received would decrease by 3 while the total number of stickers all the boys received would increase by 4. This causes the overall difference in the number of stickers between them to decrease by 7.
5.Opposite (number of boys) = 112 ÷ 7 = 16
Given that each replacement reduces the overall difference by 7 stickers, there would be 16 replacements needed to decrease the overall difference by 112 so that it reaches 14 as given in the question.
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