Math Explorer's Club


Introduction Lesson 1: Addition Lesson 2: Subtraction Lesson 3: Multiplication Lesson 4: Division Lesson 5: Calendar computations Lesson 6: Guessing a number and other tricks 
Division tipsDivision is perhaps the hardest of the elementary operation, since one does not always gets integers. For instance, 100 ÷ 23 = 4 + 8/23 (4 + 8/23 × 23 = 100). Sometimes it may be enough for our purposes to estimate the answer by rounding off. For instance, 100 ÷ 23 lies between 100 ÷ 25 = 4 and 100 ÷ 20 = 5.Simplifying the problemSuppose we want to divide a by b, when a and b have a common factor c. Then we can divide both numbers a and b by c to simplify the problem. For instance, suppose that we are trying to divide two even numbers, we can divide both by 2 to get a simpler problem. For example,Using fractionsFirst recall that dividing any number by a power of 10 is very easy: simply move the decimal point as many positions to the left as zeroes in the power of 10. For instance, 847 ÷ by 10 is just 84.7 (we moved the decimal point one place to the left) and 847 ÷ 100 = 8.47.Suppose we want to divide a number x by 25. Since 25 = 100 ÷ 4, we get that x ÷ 25 = (x × 4) ÷ 100. For instance, Now 374 × 4 is easy to do: multiply by 2 twice. 374 × 2 = 748, and 748 × 2 = 1496. Now, we divide by 100 to get 1496 ÷ 100 = 14.96. So 374 ÷ 25 = 14.96. We can use the same idea to divide by 5 = 10 ÷ 2. For instance to divide 284 by 5, we multiply by 2 and then divide by 10. We get 214 × 2 = 428, and 428 ÷ 10 = 42.8. Thus 214 ÷ 5 = 42.8. What if we are asked to divide by 0.5. This is the same as multiplying by 2, since 0.5 = 1/2. What about dividing by 0.2? Since 0.2 = 2/10 = 1/5, we can just multiply by 5. So when dividing by some decimals it may be useful to know the decimal expressions of fractions with small denominators. For instance,
Activity: Dividing with fractions
x ÷ 9, where x is a twodigit numberHere is a very simple rule to get x ÷ by 9, where x is any two digit number.
with 2 being the tens digit of 23, and 5 being the sum of the digits of 23. Now consider 78 ÷ 9. Since the sum of the digits is 15 > 9, the quotient is 8. The remainder is 7 + 8 &equiv 6 (mod 9).
Activity: Dividing by 9
Checking your answer by casting out ninesWe can use casting out nines to check the answer we have obtained. For instance, consider we try to divide 233464 by 32, and we obtain a quotient of 7295, and a remainder of 24 (which is the right answer). To verify the result we proceed as follows:
As before, if the test fails, we can be sure that we have made a mistake somewhere in our computation (or while casting out nines). If our result passes the test, we have reason to believe our answer is right. For instance, suppose we divide 1532 by 17, and we obtained a quotient of 90 and a remainder of 3. Let's see if this makes sense:
Activity: Checking your answer by casting out nines

