Try this beautiful Problem on Algebra based on Set of Fractions from AMC 10 A, 2015. You may use sequential hints to solve the problem.
Set of Fractions – AMC-10A, 2015- Problem 15
Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1,$ the value of the fraction is increased by $10 \% ?$
,
- $0$
- $1$
- $2$
- $3$
- $infinitely many$
Key Concepts
Algebra
fraction
factorization
Suggested Book | Source | Answer
Suggested Reading
Pre College Mathematics
Source of the problem
AMC-10A, 2015 Problem-15
Check the answer here, but try the problem first
$1$
Try with Hints
First Hint
According to the questation we can write $\frac{x+1}{y+1}=\frac{11 x}{10 y}$
\(\Rightarrow xy +11x-10y=0\)
\(\Rightarrow (x-10)(y-11)=-110\)
Now can you finish the problem?
Second Hint
Here \(x\) and \(y\) must positive, so $x>0$ and $y>0$, so $x-10>-10$ and $y+11>11$
Now we have to find out the factors of \(110\) and find out the possible pairs to fulfill the condition….
Now Can you finish the Problem?
Third Hint
uses the factors of $110$ , we can get the factor pairs: $(-1,110),(-2,55),$ and $(-5,22)$
But we can’t stop here because $x$ and $y$ must be relatively prime.
$(-1,110 )$ gives $x=9$ and $y=99.9$ and 99 are not relatively prime, so this doesn’t work.
$(-2,55 )$ gives $x=8$ and $y=44$. This doesn’t work.
$(-5,22)$ gives $x=5$ and $y=11$. This does work.
Therefore the one solution exist
Other useful links
- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=0qSCPw0YhUY&t=6s