Math Help

This test is based uon a remarkable property of 9. When any number is divided by 9, the remainder will equal the sum of the number’s digits, or the sum of the number’s digits after “casting out” 9′s. This may sound strange but the idea itself if easy to understand, as the following examples will show.

Example 1: 16 / 9 = 1, with 7 remainder. The digit sum of 16 (1 + 6) is also 7, the same as the remainder after dividing 16 by 9.

Example 2: 24 / 9 = 2, with 6 remainder. The digit sum of 24 (2 + 4) is also 6.

Example 3: 38 / 9 = 4, with 2 remainder. Of course, the digit sum of 38 is not 2 by 11. However, after casting out 9, such as by subracting 9 from 11, the remainder deos become 2, the same as the remainder after dividing 38 by 9.

Remainders obtained by casting out 9′s can be used to provide “check numbers” for testing your answers in addition, subtraction, multiplication, and division.

You can use any of the four methods to cast out 9′s and obtain the remainders, as shown in the following examples. The same numbers are used in each example to demonstrate that the remainders obtained are the same for all four methods.

Casting out 9′s by dividing
Divide the number by 9 to obtain the remainder:

• Numbers – Remainder
  46 – 1
  85 – 4
  198 – 0
  6,368 – 5

Casting out 9′s by adding
Add the number’s digits, and add the digits in the sum if there is more than one digit in the sum, to obtain a one-digit remainder. If the remainder is 9 or a number evenly divisible by 9 (18, 27, 36, etc), count the remainder as zero.

• Number – Remainder
  46: 4 + 6 = 10; 1 + 0 = 1
  85: 8 + 5 = 13; 1 + 3 = 4
  198: 1 + 9 + 8 = 18; 1 + 8 = 9; 0
  6,368: 6 + 3 + 6 + 8 = 23; 2 + 3 = 5

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Before working out a problem, determine how accurate your answer must be. A good estimate or approximation may be all you need and can save you a lot of needless figuring. Approximating depends upon determining the “significant figures” of the numbers involved – the figures that are important to your estimate.

For example, if an item is priced at 2 for $1.97, you can’t go far wrong in figuring that 3 will cost about $3.00. Or if you want to paint some walls measured at 2,386 square feet, you might use 2,500 in estimating how much paint you will need. If a gallon of paint covers 500 square feet, you have your answer faster than you can reaach for a pencil.

To arrive at the significant figures, “round off” a number to whatever extent may be required to suit your purpose and drop all figures (digits) to the right of this. If the first digit dropped on the right is 5 or more, add 1 to the last digit remaining; otherwise let the last digit stand as it is. The following examples will show how this is done:

• Example 1: Round off 6,437:
  To the nearest thousand: 6000
  To the nearest hundred: 6400
  To the nearest ten: 6440
• Example 2: Round off 83.652:
  To the nearest unit: 84
  To the nearest tenth: 83.7
  To the nearest hundredth: 83.65

Ways for rounding off numbers to approximate or estimate the answer in addition, multiplication, and division will be given in future posts.

In doing a problem, it’s better to think only of the results of each step and to omit any unnecessary details of the step itself. For example, when you see 36/9, think immediately of 4. Don’t go through the details of the process such as by saying “36 divided by 9 goes 4 times.” Or when adding, like 5 + 11 + 8, you just slow yourself up if you say “5 plus 11 makes 16, plus 8 makes 24.” Instead, just think “16, 24.”

Omit the words of a process as much as possible and concentrate on the results. You can work problems faster without the words.

Numbers and calculating are like the alphabet and reading. When you see a word, you don’t stop to spell out the letters that make it up; you recognize the word at once in its entirety. Nor do you stop to consider each word separately; you read words together and think of what they mean as a whole. Doing the same with numbers and calculations will enable you to solve problems faster and easier.

Combining mental and written math for best results

Short-cuts are not just a way of solving problems in your mind and in a flash, although they can help you do this. Taking the time to jot things down can often speed them up. Consider the following:

Calculating involves the mind in two processes: (1) thinking out each step in the problem, and (2) remembering the results of each step for use in succeeding steps. Naturally, the longer the problem and the more complex, the more you have to remember if you do it entirely by mental math, and the more chances there are of making a mistake.

On the other hand, if you work out the whole problem on paper, you don’t have to keep remembering the results of steps and you can concentrate on the solution. Writing it down may take longer but it reduces the likelihood of error, and this can save time in the long run.

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Short-cut methods are based upon the principle of changing “difficult” numbers and processes into easier ones. For example, to find the sum of 29 and 36, you can add 1 to 29 and subtract 1 from 36 so that the problem becomes 30 + 35; you can see at once that the answer is 65. Adding and subtracting 1 simplified the problem, but did not change the answer. This principle of “equivalency” as used in short-cuts enables you to reach the right answer faster and easier.

You can apply the same principle to multiplying. Take 28 x 15, for instance. If you halve 28 (=14) and double 15 (=30), you change the problem to 14 x 30 which, as you can see, is an easy 420 – the same answer you will get by multiplying 28 x 15.

It’s not always the size of a number that makes it difficult to handle but the KIND of number it is. You know, of course, that it’s much easier to multiply by 10 than by 9 even though 10 is the larger number. However, the “difficult” number 9 can be changed into two “easy” number, 10 minus 1, which equal 9. Therefore, to multiply a number by 9, you can multiply by 10 (simply add a zero to the number) and subtract the number, thus: 37 x 9 = 370 – 37 = 333. The 370 – 37 is, of course, 37 x (10-1) instead of 37 x 8. Either way, the answer is the same.

Incidentally, the easiest numbers to handle are 0, 1, 10 and 2, although 100, 1000, etc are also easy to get along with; so are other “zero number,” such as 20, 30, etc. If you can change a difficult number to one of these, you are on your way to an easier solution.

In many cases, you will find it faster & easier to perform two of three simple operations rather than a single more difficult one, like taking the stairs to the next floor rather than trying to make it in one big jump. To multiply 64 by 25, for instance, you can divide 64 by 4 (=16) and multiply by 100 (just add two zeros), and there’s your answer, 1,600. This works because dividing by 4 and multiplying by 100 is the as multiplying by 100/4 which equals 25. Again, the method has been simplified but the answer is not changed because 25 and 100/4 are equivalent.

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