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Begin by making your life easier: presume that there are \(\displaystyle 6\) papers on the desk. Immediately, we know that there are \(\displaystyle 2\) paper clips. Now, if there are \(\displaystyle 6\) papers, you know that there also must be \(\displaystyle 2\) greeting cards. Technically you figure this out by using the ratio:

\(\displaystyle \frac{3}{1}=\frac{6}{x}\)

By cross-multiplying you get:

\(\displaystyle 3x=6\)

Solving for \(\displaystyle x\), you clearly get \(\displaystyle x=2\).

(Many students will likely see this fact without doing the algebra, however. The numbers are rather simple.)

Now, this means that our desk has on it:

\(\displaystyle 6\) papers

\(\displaystyle 2\) paper clips

\(\displaystyle 2\) greeting cards

Therefore, you have \(\displaystyle 10\) total items.  Based on this, your ratio of paper clips to total items is:

\(\displaystyle \frac{2}{10}=\frac{1}{5}\), which is the same as \(\displaystyle 1:5\).

To begin, you need to calculate how many students are taking Old Norse. This is:

\(\displaystyle 120-63-22-27=8\)

Now, the ratio of students taking Old Norse to students taking Greek is the same thing as the fraction of students taking Old Norse to students taking Greek, or:

\(\displaystyle \frac{8}{22}\)

Next, just reduce this fraction to its lowest terms by dividing the numerator and denominator by their common factor of \(\displaystyle 2\):

\(\displaystyle \frac{4}{11}\)

This is the same as \(\displaystyle 4:11\).

To begin, you need to do a simple addition to find the total number of flowers in the garden:

\(\displaystyle 40+30+12+16=98\)

Now, the ratio of petunias to the total number of flowers in the garden can be represented by a simple division of the number of petunias by \(\displaystyle 98\). This is:

\(\displaystyle \frac{16}{98}\)

Next, reduce the fraction by dividing out the common \(\displaystyle 2\) from the numerator and the denominator:

\(\displaystyle \frac{8}{49}\)

This is the same as \(\displaystyle 8:49\).

Ratios take the form of numerator:deminator when in colon form.

\(\displaystyle \frac{4}{9}\rightarrow4:9\)

The ratio of sophomores to freshmen in the class can be expressed by the fraction \(\displaystyle \frac{6}{20}\).

Now, simplify this fraction.

\(\displaystyle \frac{6}{20}\rightarrow\frac{3}{10}\)

That fraction can also be expressed as \(\displaystyle 3:10\).

First, find out how many piano players there are in the class.

\(\displaystyle \text{Number of piano players}=25-5-6-8=6\)

Now, we can express the ratio of piano players to oboe players as the following fraction:

\(\displaystyle \frac{6}{8}=\frac{3}{4}\)

The fraction can also be expressed by \(\displaystyle 3:4\).

First, find the number of bluebonnets in the garden.

\(\displaystyle \text{Number of bluebonnets}=120-40-15-25=40\)

The ratio of bluebonnets to tulips can be expressed as a fraction:

\(\displaystyle \frac{40}{40}=\frac{1}{1}\)

The fraction can also be expressed as the ratio \(\displaystyle 1:1\).

In the colon form of a ratio, the fraction becomes numerator:denominator.

\(\displaystyle \frac{1}{10}=1:10\)

The ratio of orange juice to blueberry juice can be expressed as a fraction:

\(\displaystyle \frac{4}{2}=\frac{2}{1}\)

That fraction can also be expressed as \(\displaystyle 2:1\).