Source: NCTM Mathematics Teacher, September 2006

**Solution
**Let , and represent the ages of Bob, Jane, and Mary respectively.

Add Eq. and Eq.

——————

Add Eq. and Eq.

——————

Substitute the value of in Eq. into Eq.

Jane’s age equals .

Substitute the value of Jane’s age into Eq.

Mary’s age equals .

Substitute the value of Jane’s age into Eq.

Bob’s age equals .

**Answer**: Bob is , Jane , Mary

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**

Circumference = . Since measure of major arc , measure of minor arc . Central angle which tends arc therefore measures radians or . Triangle is isosceles, hence altitude is also the angle bisector of central angle . Triangle is a triangle with hypotenuse equal . Thus, side has length equal .

units

**Answer**: units

**Alternative solution**

Since chord is a diameter and chord is perpendicular to , arc is a mirror image of arc in the line of reflection. Points , and divide the circumference into three equal arcs each measuring . Hence, is an equilateral triangle where the circumcenter is also the centroid of the triangle.

By the Concurrency of Medians of a Triangle theorem

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**By definition of multiplication inverse

**Answer**:

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**The desirable outcomes are

where is the first number and are the second number

Total number of desirable outcomes

Number of possible outcomes

Probability that the second number is larger than the first =

**Answer**:

**Alternative solution**

First, the die is thrown and the first number comes up each with a probability equal . Then, the die is thrown again and we want the second number larger than the first.

probability =

probability =

probability =

probability =

probability =

probability =

Probability of rolling the die two times and the second number is larger than the first

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(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 such that . We have the following two equations

Multiply Eq. by and add to Eq.

————————

Divide both sides by

For to be an integer, must be a multiple of .

**Answer**: (c) only multiples of

**Alternative solution
**

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**Let represent the length of one side of a square. The following equation is true

or (extraneous solution)

The length of one side of the square equals units.

**Answer**: units

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**Let represent the number of small lemonades and the number of large lemonades sold. We have the following system of equations

Multiply Eqs. and by

Multiply Eq. by and add to Eq.

————————-

Substitute the value of into Eq.

John has sold small and large lemonades.

**Answer**: small and large

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Source: NCTM Mathematics Teacher, September 2006

**Solution**

In dollars the following equation depicts how you spent at the two stores

Collect like terms and simplify

At each store you purchase plums; in all you bought plums.

**Answer**:

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**The square bracket in the sum expression contains 1003 pairs of consecutive powers which pairwise sum to zero

**Answer**:

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Source: NCTM Mathematics Teacher, September 2006

**Solution
**Areas of pizza slices

8-inch:

10-inch:

12-inch:

14-inch:

Since is the largest fraction, you want to take a slice from the 10-inch pizza.

**Answer**: 10-inch pizza

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