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Thankfully, I have been provided 4 options:

(a) $0.0987$

(b) $0.0897$

(c) $0.0798$

(d) $0.0789$

My attempt

$$\begin{aligned} \frac{63.5\times 0.5\times 10\times 60}{2\times 96500} &= \frac{63.5\times 0.5\times 600}{2\times 96500} \\ &= \frac{63.5 \times 300}{2\times 96500} \\ &= \frac{63.5\times 150}{96500} \\ &= \frac{635\times 15}{96500} \\ &= \frac{6350+3000+150+25}{96500} \\ &= \frac{9500+25}{96500} \\ &= \frac{9525}{96500} \end{aligned}$$

I'm still stuck with a long division.

$$\require{enclose} \begin{array}{rll} 0.09 && \\[-3pt] 96500 \enclose{longdiv}{9525}\kern-.2ex \\[-3pt] \underline{868500} \\[-3pt] \end{array}$$

[I couldn't display the long division nicely.]

We do not need to continue the long division further. We can understand the answer will be (a) by checking the options.

My question

This is actually a chemistry question from a competitive exam, which involves stoichiometric calculations. I had the most trouble doing the long division. I had to guess that the number is $9$ and had to multiply $9$ by $96500$. Is there a quicker and easier way?

tryingtobeastoic
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    $$ \frac{63.5 \times 300}{2\times 96500} \ \ = \ \ \frac{63.5 \times 3 }{2\times 965 } \ \ \approx \ \ \frac{190}{2\times 965 } \ \ = \ \ \frac{95}{ 965 } \ \ , $$ which will be rather close to $ \ \frac{1}{10} \ \ . $ So I'd take choice $ \ \mathbf{(A) } \ \ $ (which a calculator confirms). –  May 17 '22 at 07:34
  • Start simplifying it: $0.5 \times 10=5$. Thus we can rewrite it as: $\dfrac {63.5 \times 5 \times 6 \times 10}{2 \times 9650 \times 10}=\dfrac {63.5 \times 5 \times 3}{9650}=\dfrac {63.5 \times 3}{965 \times 2}$ – Mauro ALLEGRANZA May 17 '22 at 07:36
  • @boojum I see. But the problem is that (b) is also pretty close to $\frac{1}{10}$, which can cause a dilemma in the exam hall. :-) – tryingtobeastoic May 17 '22 at 07:39
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    $$ \frac{95}{965} \ \ = \ \ \frac{19}{193} \ \ $$ and $$ \frac{193}{19} \ < \ 11 \ \ , $$ (in fact, it's less than $ \ 10.2 \ \ $ ) , so $ \ \frac{1}{0.0897} \ > \ \frac{1}{0.09} \ > \ 11.11 \ \ $ will not be close enough to be an issue. –  May 17 '22 at 07:46
  • Continuing on @MauroALLEGRANZA 's line it boils down to a long division of $381$ by $386$ and in a few seconds we see $9$ then $8$ emerge .. – ancient mathematician May 17 '22 at 07:47
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    If one were going to undertake the long division, the appearance of the $ \ 9 \ $ after the $ \ 0.0 \ $ in the quotient already settles the matter as far as the choices are concerned: if the $ \ 9 \ $ couldn't be managed, but an $ \ 8 \ $ could, then one would know the answer was (B). –  May 17 '22 at 08:08
  • Using the first term of the geometric series, the last result in the first comment can be continued as $$0.1·(1-0.05)·(1-0.035)^{-1}\approx0.1·(1-0.05)·(1+0.035)\approx0.1·(1-0.015)=0.0985.$$ – Lutz Lehmann May 17 '22 at 08:34
  • @RodrigodeAzevedo I was just dividing 9525 by 96500 using long division – tryingtobeastoic May 21 '22 at 08:04

2 Answers2

1

\begin{align}&\frac{63.5\times 0.5\times 10\times 60}{2\times 96500}\\&=\frac{635\times5\times3}{96500}\\ &=\frac{127\times3}{965\times4} \quad\text{(dividing by $5^2=\dfrac{100}4)$}\\&=\frac{128\times3-3}{965\times4} \quad\text{(retaining $965$ as it is near $1000)$}\\&=\frac{96-0.75}{965}\\&>\frac{95.25}{1000}\\&=0.09525.\end{align}

(a) $0.0987$
(b) $0.0897$
(c) $0.0798$
(d) $0.0789$

The only option that exceeds $0.09525$ is (a).

ryang
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1

Cancel $100$ from the denominator, and hand multiply the rest.

$$\frac{635\times3}{2\times965}=\frac{1905}{1930}\approx1$$

Therefore the answer is (a).

JMP
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