As part of a stats project, a teacher brings a bag of marbles containing six hundred
white marbles,
and four hundred red marbles. She tells the students the bag contains
one thousand total marbles, and asks her students to determine
how many red marbles are in the bag without counting them. A student randomly
draws two hundred marbles
from the bag. Of the two hundred marbles, eighty-five are red.
So based upon the sample of two hundred marbles,
the student would estimate that how many marbles in the bag were red.
Well because eighty-five out of two hundred marbles from the sample were red,
the fraction of marbles that were red were eighty-five
two hundredths. And notice how this fraction does simplify, there's a common factor of five
between eighty-five and two hundred. So if we divide
both the numerator and denominator by five, eight-five divided by five is equal to seventeen.
Two hundred divided by five is equal to forty. So if this is the fraction of red marbles from
the sample,
the student would also estimate that this fraction of marbles that would be red
from the total of one thousand marbles. Let's also write this fraction as a percentage.
To convert a fraction to a percentage, we first convert to a decimal by
dividing.
Then convert the decimal to a percentage.
So to convert seventeen fourtieths to a decimal,
we would have seventeen divided by forty, which gives us
point four, two, five. To convert decimal to a percentage,
we can move the decimal point to the right two places, and add a percent sign,
or multiply by one hundred and had a percent sign.
So seventeen fortieths
is equal to forty-two point five percent. So again as a decimal
we had zero point four, two, five,
which is equal to forty-two point five percent.
So we can also say the student would estimate that forty-two point five percent
of the total one thousand marbles would be red.
So we can pose this question two ways. We could answer the question, what is
seventeen fortieths of
one thousand? Again there are a total of one thousand marbles.
We could also answer the question, what is
forty-two
point five percent of one thousand? Let's answer both of these questions,
and then we'll also look at this a third way by setting up a proportion.
In mathematics, of means multiplication.
So in order to find seventeen fortieths of one thousand,
we want to find the product of seventeen fortieths,
and one thousand. In order to find a percent of a number,
we convert the percent to a decimal and multiply. So to find forty-two point five percent
of one thousand, because forty-two point five percent of the decimal is
zero point four, two, five. We want to find zero point four, two, five times one thousand.
To find forty-two point five percent of
one thousand. Well notice here, because we are multiplying by one thousand,
we could just move the decimal point to the right three places,
and find our product. Let's go ahead and check both of these
on the calculator. So for seventeen fortieths times one thousand,
I'd probably put the fraction in parenthesis.
So there's seventeen fortieths, and then times one thousand,
which is equal to four hundred twenty-five.
And notice how point four, two, five
times one thousand will give us the same result,
four hundred twenty-five.
So this would be the students estimate for the total number of red marbles,
out of the one thousand marbles.
Notice how this is actually just an estimate because we already know,
or at least the teacher knows, there are only four hundred red marbles
out of the one thousand. Now let's take a look at this a third way,
by setting up a proportion. A proportion is formed when we set two ratios, or rates equal to each
other.
So for this first ratio using the sample,
we know eighty-five out of two hundred marbles are red,
so here we're comparing the red marbles to the sample size,
and we'd estimate this would be equal to the number of red marbles
out of the total of one thousand marbles. So the second ratio would be
r, the unknown red marbles, to the total of one thousand marbles.
So this is our proportion, and now to solve the proportion,
we would cross multiply, and solve for r. So to cross multiply we'd have two hundred
times r
must equal eighty-five times one thousand. Well,
two hundred times r is two hundred r, equals
eighty-five times one thousand would be eighty-five thousand.
And then to solve for r we would divide both sides by two hundred.
Simplifying we have r equals
eighty-five thousand divided by two hundred is,
again four hundred twenty-five.
So as you can see there are several ways to solve
this type of problem. I hope you found this helpful.