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For example when diving 105 / 148. After you add a number 0 to the numerator, the division becomes 1050 / 148.

The answer becomes a decimal with 1050 / 148. The two numbers are not divisible by a common number so the first step i have to do is guess how many times 148 goes into 1050.

My approach is to round 148 to 200 and since 200 * 5 = 1000 the first number in the quotient must be above 5. I then had to guess 148 * 5, then 148 * 6, then 148 * 7. I'm studying for a standardized test that does not allow calculators or else i would just use a calculator.

Is there a better or smarter strategy to guessing how many times 148 goes into 1050 other than guessing like i did?

Ian
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  • The typical approach is to use $ln$ function. Since you have no tools to help you, maybe this manual method could help:http://calculus-geometry.hubpages.com/hub/How-to-Estimate-Natural-Log-By-Hand – NoChance Aug 23 '15 at 06:35
  • If you need lots of decimal places, it is worthwhile just writing out all multiples of $148$ from $2 \times 148$ to $9 \times 148$ before you start. – TonyK Aug 23 '15 at 06:47
  • Once you have a guess, you can always subtract it off and look at the remainder. e.g. if I guess $3$ goes into $100$ 30 times, the remainder is $100 - 3 \cdot 30 = 10$, and if I want can improve my estimate by seeing how many times $3$ goes into the remainder. –  Aug 23 '15 at 10:31
  • @Hurkyl I really don't understand the concept in your example. Can you give one or two more examples? – Ian Aug 23 '15 at 12:05

2 Answers2

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For this particular problem:

I recognize $105$ as $3 \times 5 \times 7$.

Then I wonder: Are any of these factors shared by $148$?

No, unfortunately not: We can quickly see the latter is not divisible by $3$ or $5$.

However, its predecessor $147 = 7 \times 21 = 3 \times 7 \times 7$.

So: I might just estimate by replacing the denominator:

$$\frac{105}{148} \approx \frac{105}{147} = \frac{3 \times 5 \times 7}{3 \times 7 \times 7} = \frac{5}{7}$$

If you happen to know that $\frac{1}{7} = 0.\overline{142857}$, then you might recognize $\frac{5}{7}$ as just over $0.71$.

Finally: Since we decreased the denominator, the original ratio is a bit less than our adjusted one. Since our adjusted ratio is just over $0.71$, this seems like a pretty good guess.

(Indeed: $\frac{105}{148} = 0.709459\ldots$)

Benjamin Dickman
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The strategy depends on the experience someone has. I would round up 148 to 150. And then divide 1050 by 150. The result is 7. Therefore $\frac{105}{148}\approx 0.7$. To round up 148 to 200 is too imprecise, but calculating with 1050 instead of 105 was good idea.

Another possibility is to divide 1000 (rounded down) by 150 (rounded up). The fraction becomes $\frac{1000}{150}=\frac{100}{15}=\frac{20}{3}\approx 7$, because $21/3=7$. Dividing the result by 10 is 0.7.

callculus42
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  • That's a neat idea. Rounding up and down both of numbers and trying to find a common denominator. – Ian Aug 23 '15 at 07:44