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.Tofigure out the tens digit of the two-digit answer, you need to askhow many times 23 goes into 680.If you try 30, you ll see thatit s slightly too much, as 30 23 690.Now you know thatthe answer is 20 something, and you can say so.Then subtract23 20 460 from 682 to obtain 222.Since 23 9 207, theanswer is 29 with a remainder of 222 207 15.2923 6 284602222071515Answer: 2923Now consider:491 62Since 491 is less than 62 10 620, your answer will sim-ply be a one-digit number with a remainder.You might guess 8,but 62 8 496, which is a little high.Since 62 7 434, the5answer is 7 with a remainder of 491 434 57, or 7 7.627962 4 143457 Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 89Divide and Conquer: Mental Division 8957Answer: 762Actually, there s a nifty trick to make problems like this eas-ier.Remember how you first tried multiplying 62 8, butfound it came out a little high at 496? Well, that wasn t awasted effort.Aside from knowing that the answer is 7, youcan also compute the remainder right away.Since 496 is 5 morethan 491, the remainder will be 5 less than 62, the divisor.5Since 62 5 57, your answer is 7 7.The reason this trick62works is because 491 (62 8) 5 62 (7 1) 5 (627 62) 5 (62 7) (62 5) 62 7 57.Now try 380 39 using the shortcut we just learned.So 3910 390, which is too high by 10.Hence the answer is 9 with aremainder of 39 10 29.Your next challenge is to divide a two-digit number into afour-digit number:3657 54Since 54 100 5400, you know your answer will be atwo-digit number.To arrive at the first digit of the answer, youneed to figure how many times 54 goes into 3657.Since 5470 3780 is a little high, you know the answer must be 60something.Next, multiply 54 60 3240 and subtract 3657 3240417.Once you say the 60, your problem has been simplified to417 54.Since 54 8 432 is a little too high, your last digitis 7 with the remainder 54 15 39. Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 9090 Secrets of Mental Math6754 3 56 732404173783939Answer: 6754Now try your hand at a problem with a three-digit answer:9467 137284 713 9 6910036726010710433Answer: 72813Simplifying Division ProblemsIf by this point you re suffering from brain strain, relax.Aspromised, I want to share with you a couple of tricks for simpli-fying certain mental division problems.These tricks are basedon the principle of dividing both parts of the problem by a com-mon factor.If both numbers in the problem are even numbers,you can make the problem twice as easy by dividing each num- Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 91Divide and Conquer: Mental Division 91ber by 2 before you begin.For example, 858 16 has two evennumbers, and dividing each by 2 yields the much simpler prob-lem of 429 8:53 Divide by 2 535 216 8 8 8 4 9800 40058 2948 2410 510 5Answer: 53 Answer: 5316 8As you can see, the remainders 10 and 5 are not the same; butif you write the remainder in the form of a fraction, you get1 5that 0 is the same as.Therefore, when using this method, you16 8must always express the answer in fractional form.We ve done both sets of calculations for you to see how mucheasier it is.Now you try one for practice:3618 5467 Divide by 2 676 8 8 954 3 1 27 1 03240 1620378 189378 1890 0Answer: 67The problem on the right is much easier to calculate men-tally.If you re really alert, you could divide both sides of the Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 9292 Secrets of Mental Mathoriginal problem by 18 to arrive at an even simpler problem:201 3 67.Watch for problems that can be divided by 2 twice, such as1652 36:4511652 36 826 18 413 9 9 4 32 2360534588Answer: 459I usually find it easier to divide the problem by 2 twice thanto divide both numbers by 4.Next, when both numbers end in0, you can divide each by 10:887 5580 70 58 7105622Answer: 87But if both numbers end in 5, double them and then divideboth by 10 to simplify the problem.For example:13475 35 950 70 95 77 952 107025214 Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 93Divide and Conquer: Mental Division 934Answer: 137Finally, if the divisor ends in 5 and the number you re divid-ing into ends in 0, multiply both by 2 and then divide by 10, justas you did above:199 1 87890 45 1780 90 178 92 1090888177Answer: 199EXERCISE: TWO-DIGIT DIVISIONHere you ll find a variety of two-digit division problems thatwill test your mental prowess, using the simplification tech-niques explained earlier in this chapter.Check the end of thebook for answers and explanations.1.2.3.738 17 591 24 321 794.5.6.4268 28 7214 11 3074 18MATCHING WITS WITH A CALCULATOR:LEARNING DECIMALIZATIONAs you might guess, I like to work some magic when I convertfractions to decimals.In the case of one-digit fractions, the best Benj_0307338401_4p_c04_r1.r.qxd 5/4/06 1:43 PM Page 9494 Secrets of Mental Mathway is to commit the following fractions from halves throughelevenths to memory.This isn t as hard as it sounds.As you llsee below, most one-digit fractions have special properties thatmake them hard to forget
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