What's The Best Strategy To Solve This?

Number Theory Level 2

What is the smallest fraction, $\dfrac{a}{b},$ that is greater than $\dfrac{3}{4}$ if $a$ and $b$ must both be integers $\leq 10 \ ?$

If $\dfrac{a}{b}$ is your solution, then submit your answer as $a + b$ . (For example, if your solution is $\dfrac{2}{3},$ type in 5)

There's a way to figure this out very quickly!

The answer is 16.

1 solution

Zandra Vinegar Staff
Mar 9, 2016

The smallest fraction greater than $\frac{3}{4}$ with numerator and denominator less than or equal to 10 is $\frac{7}{9}.$ Therefore, the value to submit as your solution is $7+9 = \fbox{16}.$

Solving this kind of problem quickly is all about understanding what fractions might be reasonable answers and why.

Given the constraint that $a$ and $b$ are both less than or equal to 10, there's a table of 100 potential fractions! Our goal is to actually evaluate as few of these as possible in the process of finding the one closest to and greater than $\frac{3}{4} = .75$ There's actually a strategy that works with this fraction and many others which yields a solution extremely quickly!

A first observation is that, considering the set of potential fractions $\frac{a}{b}$ which share a common denominator, $b$ we can find the term closest to but greater than $\frac{3}{4}$ by scaling $\frac{.75}{1}$ to also have that same denominator. For example, considering sevenths ( $b=7$ ), we can scale $\frac{.75}{1}$ to $\frac{7 \times .75}{7} = \frac{5.25}{7}.$ Therefore, fraction with denominator 7 that is greater than $\frac{3}{4}$ but as close to $\frac{3}{4}$ as possible is $\frac{6}{7}$ . And the difference is now easy to calculate because the fractions have a common denominator: $\frac{6}{7} - \frac{5.25}{7} = \frac{.25}{7}$

The results: First scale .75, then round up or jump to the next integer if the given one is already whole, and then find the difference:

.75 Scaled	Jump up (same denominator)	Difference
$\frac{.75}{1}$	$\frac{1}{1}$	$\frac{.25}{1}$
$\frac{1.5}{2}$	$\frac{2}{2}$	$\frac{.5}{2}$
$\frac{2.25}{3}$	$\frac{3}{3}$	$\frac{.75}{3}$
$\frac{3}{4}$	$\frac{4}{4}$	$\frac{1}{4}$
$\frac{3.75}{5}$	$\frac{4}{5}$	$\frac{.25}{5}$
$\frac{4.5}{6}$	$\frac{5}{6}$	$\frac{.5}{6}$
$\frac{5.25}{7}$	$\frac{6}{7}$	$\frac{.75}{7}$
$\frac{6}{8}$	$\frac{7}{8}$	$\frac{1}{8}$
$\frac{6.75}{9}$	$\frac{7}{9}$	$\frac{.25}{9}$
$\frac{7.5}{10}$	$\frac{8}{10}$	$\frac{.5}{10}$

See the pattern? The numerator cycles through the pattern {.25, .5, .75, 1...} as the denominator grows. Here, the solution will be the occurrence of the smallest numerator, .25, over the largest denominator that it appears with, 9. Therefore, the answer is $\frac{7}{9},$ and the value to submit as your solution is $7+9 = 16.$

Bonus:

The strategy here can be applied faster and with other fractions if you're comfortable with it, as long as you can recognize what cyclic pattern the decimals of the multiples will have. For example, consider the multiples of $\frac{4}{7}:$ And, this time, say that the numerator & denominator can be any integer less than or equal to 20. Find the pattern, and then find the fraction which is as close to $\frac{4}{7}$ as possible while being greater than $\frac{4}{7}$ and having integer numerator and denominator less than or equal to 20.

$\frac{4}{7} \times 1$	$= .571428\overline{571428}$
$\frac{4}{7} \times 2$	$= 1.142857\overline{142857}$
$\frac{4}{7} \times 3$	$= 1.714285\overline{714285}$
$\frac{4}{7} \times 4$	$= 2.285714\overline{285714}$
$\frac{4}{7} \times 5$	$= 2.857142\overline{857142}$
$\frac{4}{7} \times 6$	$= 3.428571\overline{428571}$
$\frac{4}{7} \times 7$	$= 4$
$\frac{4}{7} \times 8$	$= 4.571428\overline{571428}$
$\frac{4}{7} \times 9$	$= 5.142857\overline{142857}$
$\frac{4}{7} \times 10$	$= 5.714285\overline{714285}$
$\frac{4}{7} \times 11$	$= 6.285714\overline{285714}$
$\frac{4}{7} \times 12$	$= 6.857142\overline{857142}$
$\frac{4}{7} \times 13$	$= 7.428571\overline{428571}$
$\frac{4}{7} \times 14$	$= 8$
...	...

The hint is misleading. It is best not to include it.

Reginald Micu - 5 years, 3 months ago

My intent was that that is one strategy for ruling out large swaths of fractions. But perhaps you're right, I'm going to remove it.

Zandra Vinegar Staff - 5 years, 3 months ago

what is the answer for your bonus question mentioned above???

Is it (9/15)..............????

please reply asap.....

Rajat Pathak - 5 years ago

Why 4/5 is not correct answer?? What's wrong with that fraction??? And they asked for smallest fraction

Kumar Patchakanthala - 5 years, 3 months ago

4/5 = 0.8, but 7/9 = 0.77778, which is smaller than 0.8.

Edgar Felizmenio - 5 years, 3 months ago

What's The Best Strategy To Solve This?

The answer is 16.

1 solution

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