Fraction Cross Out

Probability Level 3

A student is simplifying fractions, and notices that since 16 64 = 1 4 \frac{16}{64} = \frac{1}{4} , he can “cross out” the sixes as a shortcut.

Obviously, this is an incorrect method, but what are all of the fractions that this supposed shortcut works for? Include all fractions a b \frac{a}{b} where a < b a < b , a a has 2 2 unique digits, b b has 2 2 unique digits, and the second digit of a a is the same as the first digit of b b so that they cross out, and enter the sum of all the possible a a and b b values as your answer.


The answer is 432.

David Vreken by David Vreken , 42, USA

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1 solution

Jeremy Galvagni
Aug 27, 2018

Write the equation as 10 x + y 10 y + z = x z \frac{10x+y}{10y+z}=\frac{x}{z}

Cross multiply and move some terms then refactor to rewrite as 10 x ( y − z ) = z ( y − x ) 10x(y-z)=z(y-x) .

This is significant because there must be a factor of 5 5 on the right which means either z = 5 z=5 or y − x = 5 y-x=5 .

Case 1: z = 5 z=5

The equation becomes 10 x ( y − 5 ) = 5 ( y − x ) 10x(y-5)=5(y-x) . Expand and solve for y y to get y = 9 x 2 x − 1 y=\frac{9x}{2x-1} .

This has three positive integer solutions of which one ( x = y = z = 5 ) x=y=z=5) doesn't count. The good ones are x = 1 , y = 9 x=1, y=9 and x = 2 , y = 6 x=2, y=6 and we have the fractions

19 95 = 1 5 \frac{19}{95}=\frac{1}{5} and 26 65 = 2 5 \frac{26}{65}=\frac{2}{5}

Case 2: y − x = 5 y-x=5

Substituting y = x + 5 y=x+5 the equation becomes 10 x ( x + 5 − z ) = z ( 5 ) 10x(x+5-z)=z(5) . Expand and solve for z z to get z = 2 x 2 + 10 x 2 x + 1 z=\frac{2x^{2}+10x}{2x+1}

This has two positive integer solutions which both work: x = 1 , z = 4 x=1, z=4 and x = 4 , z = 8 x=4, z=8 and we have fractions

16 64 = 1 4 \frac{16}{64}=\frac{1}{4} and 49 98 = 4 8 \frac{49}{98}=\frac{4}{8}

The solution to this problem is then 19 + 95 + 26 + 65 + 16 + 64 + 49 + 98 = 432 19+95+26+65+16+64+49+98=\boxed{432}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Great solution!

David Vreken - 2 years, 9 months ago

Would it be valid to say that 49 98 = 4 8 \frac{49}{98}=\frac{4}{8} is the fully simplified form? If the student is simplifying fractions, then his teacher would mark points off for him not getting the correct answer.

Blan Morrison - 2 years, 8 months ago

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Of course, this is not fully simplified. It's not really simplified at all, but they are equivalent fractions. I couldn't imagine a student reducing 49 to 4 and 98 to 8 as the ratio is not a whole number.

By the wording of the problem, 49 and 98 should be included in the sum.

Jeremy Galvagni - 2 years, 8 months ago

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