# Tag Archives: fraction

## Sums of Reciprocals

The sum of the positive divisors of is . Find the sum of the reciprocals of the positive divisors of . Source: NCTM Mathematics Teacher, February 2008 Solution Given that , the number of divisors is . The first twelve … Continue reading

## The Second Number

The sum of three numbers is . The first number is of the second, and the second is of the third. What is the second number? Source: NCTM Mathematics Teacher, February 2008 Solution Let be the first, second, and third … Continue reading

## Partial Fractions

When is decomposed into partial fractions (fractions that sum to the given fraction), what is the sum of the numerators when each fraction is reduced to lowest terms? Source: NCTM Mathematics Teacher, August 2006 Solution Solving for Collecting like terms … Continue reading

## Values of Integers

If and are integers such that , what are the possible values of ? (a) only (b) only (c) only multiples of (d) any integer (e) none of these Source: NCTM Mathematics Teacher, September 2006 Solution Let be an integer … Continue reading

## Red and Yellow Plums

One store sold red plums at for and yellow plums at for . A second store sold red plums at for and yellow plums at for . You bought red plums and yellow plums from each store, spending a total … Continue reading

## Biggest Pizza Slice

An 8-inch pizza is cut into equal slices. A 10-inch pizza is cut into equal slices. A 12-inch pizza is cut into equal slices. A 14-inch pizza is cut into equal slices. From which pizza would you take a slice … Continue reading

## Integral Points

A point is called integral if both and are integers. How many points on the graph of are integral points? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Divide by , Divide by Divide by , no solution , duplicate solution … Continue reading

## Passing Tests

A group of boys and girls took a test. Exactly of the boys and exactly of the girls passed the test. If an equal number of boys and girls passed the test, what fraction of the entire group passed? Source: … Continue reading

## Funny Face

The radius of each small circle is of the radius of the large surrounding circle. The radius of the middle-sized circle is twice that of either small circle. What fraction of the area of the large circle is not shaded? … Continue reading

## Divide in Half

Divide in Half How many ordered pairs of integers satisfy ? Source: NCTM Mathematics Teacher 2006 SOLUTION For to be an integer must divide evenly. ordered pairs satisfy the requirement: , and . But, I don’t know how to prove … Continue reading