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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.
There are several ways to speed up computations done by hand:
Memorizing numerical facts:
Here is a list of facts that have to be memorized:
The first 11 powers of 2:
The prime numbers smaller than 110: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 109.
The number of primes smaller than 100: there are 25 primes smaller than 100.
The perfect squares of numbers from 1 to 20: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
The factorials from 1 to 6:
Some Pythagorean triples (remember that all the scalings of these triplets are also Pythagorean and practice how to recognize them):
The perfect squares/square roots from 1 to 400:
Also, note that:
A few irrational combinations with approximation of 1 decimal:
As you solve a problem do not perform all the intermediate computations as you go. At least, not always: sometimes it is impossible to proceed without computing a partial result.
However, in many cases, we can simply postpone all the computations until the final step. Then, we will be able to apply simplifications and estimations to derive the final result much quicker.
Consider that in many cases, you perform multiplications only to divide by the same numbers later on. While this is relatively painless if you have enough time and a calculator, it becomes a hindrance when you have limited time and the calculator is not allowed.
In a period of 150 days a trader made $ 25 a day for 60 days and lost $5 a day for 90 days. What was his average profit/loss?
A 'traditional' way of solving goes from step to step like this:
We compute the total gain: 25 x 60 = 1500.
We compute the total loss: 5 x 90 = 450.
We compute the net gain: 1500 - 450 = 1050.
We compute the average daily gain: 1050 / 150 = 7.
However, it is much faster to postpone all the partial computations and cumulate them all in an expression as in the following solution:
It is now easy to see that we can simplify the fraction by 30:
This fraction can be simplified further by 5:
Postponing the partial computations not only shortens the execution time but also helps with accuracy since it keeps the numbers small. Especially when a calculator is not allowed, keeping the numbers small is a computational advantage.
There are many, many shortcuts and we will not go into describing them all. More importantly, it is beneficial for each student to figure out and use their own set of shortcuts.
First of all we make as many pairs of 2 and 5 as possible, each pair will multiply to 10 and become a 0 at the end of the number. So we can start writing out the result from the end. In this case, we can form 2 pairs, therefore the result will end with 00.
Then, we look at the remaining factors:
and we notice that 3 can be written as 2 + 1:
Since we have already memorized the powers of 2 we can use them directly:
Many, many such tricks can be devised and we leave it to the ingenuity of the student to do so. Keep in mind that these tricks involve the replacement of costly multiplications and divisions by convenient additions and subtractions.
Estimating is very handy when a set of answer choices is already provided and exact computation would not bring any additional benefit.
These techniques require active practice. Do not plug these expressions in the calculator! Instead, try to find the shortest, most elegant way of simplifying operations or translating them into simpler operations.