Whenever we make a calculation, we should consider whether or not the answer makes sense.

EXAMPLE: Nick bought a magazine for $2 and gave the clerk a $10 bill. How much change should Nick get back? Suppose that your answer is $3. Does that make sense?

You don't have to know the addition facts to feel that Nick is getting cheated. The answer doesn't make sense.

We should always check our calculations and other people's calculations if our money is at stake. Since we aren't always able to check answers on paper, we should be able to check them with mental arithmetic. Here are some easy mental arithmetic shortcuts:

• To multiply a number by 10, just add one zero: 25 x 10 = 250. How would you multiply 25 by 100?
  
• To divide a number 10, move the decimal point one place to the left. So 88 ÷ 10 = 8.8. What is 88 divided by 100?
  
• In doing some additions move part of one addend to another. So in 23 + 17 move the 3 in 23 over to the 17 so that you have 20 + 20.
  
• In multiplication split one factor into 10 and the odd number and multiply both by the other factor. So 12 x 45 = (10 + 2) x 45. 10 x 45 = 450 and 2 x 45 = 90 and 450 + 90 = 440 + 100 = 540.

CHALLENGE 1: Check these products by mental arithmetic:

3 x 12 = 38 , 5 x 11 = 55 , 11 x 15 = 175

EXAMPLE 1: Ted bought items costing $1.35, 3.97, and 4.76. The bill was $12.08. How can he make a quick estimate to see if the sum was reasonable?

Since he only needs an estimate, he can use rounding. He can round off the amounts to the nearest dollar: $1, 4, and 5. His estimated sum is $10, so $12.08 is not reasonable. The correct answer is $10.08.

CHALLENGE 2: Mentally estimate of the sum of 1.76 + 1.12 + 1.95.

EXAMPLE 2: Tom used his calculator to multiply 23 times 47. The answer he got was 1,771. Was his answer reasonable?

He could round the 23 to 20 and the 47 to 50 and then mentally multiply 20 times 50 (multiply 2 x 5 = 10 and add two zeroes, to get 1,000. His calculator answer was not reasonable. He had entered 77 instead of 47. The correct answer is 1,081. Notice that you can't switch numbers between factors in multiplication in the same way you can between addends in addition. In multiplying 23 x 47 you can't accurately move the 3 from 23 to the 47 so that you have 20 x 50. Notice that 23 x 47 = 1,081 and 20 x 50 = 1,000. However, this is close enough to make a useful estimate.

CHALLENGE 3: Estimate the product of 31 x 69.

EXAMPLE 3: Jake multiplied 30 by 50 and got 150.

The correct product is 1500. When you test for reasonableness, always check the order of magnitude. Jake's answer is off by one order of magnitude. You might check this case by changing 30 x 50 to 3 x 500. It is easy to see what the order of magnitude should be.

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