Multiplication Tricks

- To multiply any number of two figures by 11: Write the sum of the figures between them. Thus, \(34 \times 11 = 374\). (\(3 + 4 = 7\); place the seven between the 3 and 4) When the sum is of the figures is more than 9, increase the left-hand number by the 1 to carry. Thus, \(98 \times 11 = 1078\) (\(9 + 8 = 17\); add 1 to 9 to get 10; place the seven between the 10 and 8)
- To square any number of 9s immediately without multiplying: set down as many 9s less one (beginning at the left) as there are 9s in the given number, an 8, as many 0s as you do 9s, and a 1. Example: \(999^{2}\): put down 2 9s: 99. Then an 8: 998. then 2 0s (because you have 2 9s): 99800. Then append a 1: 998001.
- To square any number ending in 5: Omit the 5, and multiply the number as it will then stand by the next higher number, and append 25 to the product. Example: \(35^{2}\). Omit the 5: 3. Next multiply 3 by the next higher number, 4: \(3 \times 4 = 12\). Finally, append 25: 1225. That's the answer: 1225.
- To square any compound fraction containing \( \frac{1}{2}\), like \(5 \frac{1}{2}\) for instance, Multiply the whole number by the next higher whole number and append \( \frac{1}{4}\) to the product. Thus, \( 5 \frac{1}{2} \times 5 \frac{1}{2} = 30 \frac{1}{4}\). ( \( 5 \times (5+1) = 30 \); tag on \( \frac{1}{4}\) to get \(30 \frac{1}{4}.\) )
- To multiply any two like numbers with fractions that sum to 1 (for instance, \( 4 \frac{3}{4} \times 4 \frac{1}{4} \) ), multiply the whole number by the next highest number (\(4 \times 5\)) and append the product of the fractions (\(\frac{3}{4} \times \frac{1}{4}\)). In the case of \(4 \frac{3}{4} \times 4 \frac{1}{4}\), \(4 \times 5 = 20 \). Then append the product of \(\frac{3}{4} \times \frac{1}{4}\), \(\frac{3}{16}\). Thus, \(20 \frac{3}{16}\).
- To multiply any two numbers whose ones digits sum to 10 and with like remaining numbers (for instance, \(106 \times 104) \) multiply the upper tens numbers by the next higher number (in this case, \(10 \times 11\)) and multiply the ones digits that sum to 10 (\(6 \times 4\)) and then set the products next to one another successively (11024). Another example is \(57 \times 53\). \(5 \times 6 = 30\); \(7 \times 3 = 21\); answer is 3021.
- To multiply any number by any number of 9's (for instance, \(28 \times 99)\), append as many 0's to the multiplicand as there 9's in the multiplier (2800), and from this number subtract the multiplicand (\(2800 - 28 = 2772\)). The remainder is the answer. (2772)
- Cross multiplication is a method of multiplying large numbers in a single line. Take the example \(18 \times 76\). First multiply \(8 \times 6\), set down the 8, and carry the 4. Next multiply \(1 \times 6\), \(7 \times 8\), add the products and add the carried 4 to give 66. Set down the 6, multiply \(1 \times 7\), and add 6 to the product to give 13 which you set down to conclude the problem and yield the answer, 1368. (To learn this, work a few examples on paper.)
- Here's a rather complicated trick for calculating any two 2-digit numbers:
- Calculate the average of the two numbers.
- Square your answer (here's where you need to know the squares of numbers through 100)
- Subtract the largest of your original two numbers from the average
- Square the answer
- Subtract it from your answer in step 2 for the final answer

Note: To do this trick, you need to know the squares of numbers up to 100 in advance.

Example: \(27 \times 15\)

First, calculate the average (\( \frac{42}{2} = 21\))

Then square it (\(21^{2} = 441)\)

Now subtract largest number from the average (\(27 - 21 = 6\))

Square your answer (\(6^{2} = 36)\)

Subtract the square from the first square (\(441 - 36 = 405\)). And there's your answer: 405

This works every time, but as the average and differences can end in .5, you have to know these squares as well.

By the way, this trick works because \( (x + y)(x - y) = x^{2} - y^{2} \)

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