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AMC 10 Geometry Math Olympiad USA Math Olympiad

Problem on Fraction | AMC 10A, 2015 | Question 15

Try this beautiful Problem on Fraction from Algebra from AMC 10A, 2015.

Fraction – 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 $10 \%$ ?

,

  • $0$
  • $1$
  • $2$
  • $3$
  • \(4\)

Key Concepts


algebra

Fraction

Check the Answer


Answer: $1$

AMC-10A (2015) Problem 15

Pre College Mathematics

Try with Hints


Given that $\frac{x}{y},$ is a fraction where $x$ and $y$ are relatively prime positive integers. We have to find out the numbers of fraction if both numerator and denominator are increased by 1.

According to the question we have $\frac{x+1}{y+1}=\frac{11 x}{10 y}$

Can you now finish the problem ……….

Now from the equation we can say that $x+1>\frac{11}{10} \cdot x$ so $x$ is at most 9
By multiplying by $\frac{y+1}{x}$ and simplifying we can rewrite the condition as $y=\frac{11 x}{10-x}$. since $x$ and $y$ are integer, this only has solutions for $x \in{5,8,9} .$ However, only the first yields a $y$ that is relative prime to $x$

can you finish the problem……..

Therefore the Possible answer will be \(1\)

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Categories
Algebra AMC-8 Math Olympiad

Mixture | Algebra | AMC 8, 2002 | Problem 24

Try this beautiful problem from Algebra based on Mixture from AMC-8, 2002.

Mixture | AMC-8, 2002 | Problem 24


Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?

  • 34%
  • 40%
  • 26%

Key Concepts


Algebra

Mixture

Percentage

Check the Answer


Answer:40%

AMC-8, 2002 problem 24

Challenges and Thrills in Pre College Mathematics

Try with Hints


Find the amount of juice that a pear and a orange can gives…

Can you now finish the problem ……….

Find total mixture

can you finish the problem……..

3 pear gives 8 ounces of juice .

A pear gives \(\frac {8}{3}\) ounces of juice per pear

2 orange gives 8 ounces of juice per orange

An orange gives \(\frac {8}{2}\)=4 ounces of juice per orange.

Therefore the total mixer =\({\frac{8}{3}+4}\)

If She makes a pear-orange juice blend from an equal number of pears and oranges then percent of the blend is pear juice= \(\frac{\frac{8}{3}}{\frac{8}{3}+4} \times 100 =40\)

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