6th Grade Math Textbook, Fundamentals

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Lesson 7-15 for exercise sets.

Problem 2: Suppose you are given a jar containing 6 blue marbles

and 2 red marbles. So , or 75%, of the marbles are blue.

You are challenged to reduce the percent of blue marbles in the jar
to 20% by adding more blue and red marbles. The only restriction is
that each time you add marbles, you must double the total number
of marbles in the jar. Can this be done?

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8

Read to understand what is being asked.

List the facts and restate the question.

Facts: A jar holds 8 marbles—6 blue and 2 red.

So 75% of the marbles are blue.

You can add blue and red marbles to the jar.

Whenever you add any marbles, you must

double the total number of marbles in the jar.

Question:Can you reduce the percent of blue marbles

in the jar to 20%?

Select a strategy.

You could try to solve this problem by exploring what happens

when you add different numbers of red marbles and blue marbles

to the jar. However, this could take a very long time, and it might

not lead to the answer. Since it is not immediately obvious how

to approach this problem, you might first try to reason logically

about what you are being asked to do.

Apply the strategy.

You are starting with a jar that contains exactly 8 marbles. The first

time you double the number of marbles, there will be 16 marbles in

the jar. The second time you double the number of marbles, there will

be 32 marbles in the jar. You know that 8, 16, and 32 are all powers

of 2. Specifically, 8  2   2   2  23 , 16  2     2   2   2  24 , and

32  2  2   2   2   2  25. In fact, since you must always double

the number of marbles in the jar, the number of marbles in the jar

will always be a power of 2.

Now 20% is equivalent to , so you would need of the marbles

to be blue. This would mean that the number of marbles in the

jar would have to be divisible by 5. However, a number that is a

power of 2 is not divisible by 5 because its only prime factor is 2.

So no matter how many blue and red marbles you add to the jar,

it is impossible to double the total number of marbles and yet

reduce the percent of blue marbles in the jar to 20%.

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Check to make sure your answer makes sense.

Using a calculator, you can try taking 20% of various powers

of 2. Each and every time, you will discover that the result is

not a whole number.