Surprisingly, we haven't done much on this blog with decimals, except for converting them back and forth between (and comparing them to) fractions. And that was 4+ months ago. Today we will multiply with decimals, with a little bit of rounding thrown in. This concept appeared 7 times last year, with a median placement at question # 60.
Number Dojo Level: 197
When we are working on any estimation problems, the biggest key is to decide what to round to (and how much to round). Keep in mind that your answer must be within 5% of the exact answer (above or below) to be considered correct. As long as you are rounding to a number less than 5% away from your original number, your answer should be in range. It is even better if you can round one number up and another one down.
Sometimes, you may be asked to multiply two or more small decimals together, each of which is less than 20. Two of the numbers may be close to (and surrounding) a central number which is an integer. In these cases, you can round both of them to that integer, and then square the integer. Then simply round and multiply that result by the third number.
Example 1: 5.3 x 4.7 x 1.3
- Notice that 5.3 and 4.7 are close to (and surround) 5. Round both to 5 and square it to get 25.
- Now multiply 25 by 1.3 (using Mult25) to get 32.5. Since your answer has to be an integer, round it to either 32 or 33.
- The exact answer would be 32.383, so your acceptable (+/- 5%) range would be from 31 to 34. Either way you're fine.
Example 2: 6.8 x 7.2 x 2.2
- Notice that 6.8 and 7.2 are close to (and surround) 7. Round both to 7 and square it to get 49.
- Now multiply 49 by 2.2. (Use Double/Half to make it 98 x 1.1. Then, using Mult11, you get 107.8. Round it to 108.
- The exact answer would be 107.712, so your acceptable (+/- 5%) range would be from 103 to 113. You are safely in the middle of the range.
Example 3: 5.8 x 6.2 x 249
- Notice that 5.8 and 6.2 are close to (and surround) 6. Round both to 6 and square it to get 36.
- Now round 249 to 250 and multiply 36 by 250 (again using Mult25). The answer is 9000.
- The exact answer would be 8954.04, so your acceptable (+/- 5%) range is from 8507 to 9401.
Other times, you will be given a decimal which is a fractional equivalent of a common fraction (such as .3333 for 1/3 or .125 for 1/8). In these cases, it is best to convert the decimal to that fraction and then multiply.
Example 4: 166.667 x 981
- Notice that 166.667 is almost exactly 1/6 of 1000. So change this problem to 981 x 1/6 x 1000.
- Multiplying 981 by 1/6 is the same as dividing 981 by 6. You get 163.5.
- Now multiply by 1000 to get 163500.
- The exact answer would be 163500.327, so we are obviously within acceptable range (from 155326 to 171675).
Example 5: 0.375 x 241 x 459
- (This one was on the state test!) Notice that 0.375 is the same as 3/8.
- 241 can be rounded to 240, which is a multiple of 8. Think: 3/8 x 240 x 460, or 240 ÷ 8 x 3 x 460. (Remember to divide before you multiply!)
- 240 ÷ 8 is 30. 30 x 3 is 90. We are left with 90 x 460. Write down 00 and worry about just 9 x 46.
- 9 x 46 is the same as (10 - 1) x 46, which is 460 - 46. Write down 414 in front of the 00; your answer is 41400.
- The exact answer is 41482.125, so we are safely within acceptable range (from 39409 to 43556).