Write the Following Ratio Using Two Other Notations: Fraction, Colon, and Word Form

A worksheet gives one instruction: write the following ratio using two other notations. Three classmates answer $12 : 7$, $\tfrac{12}{7}$, and “twelve to seven” - and every one of them is right. You saw the same trick in the picture: three blue squares stand next to one yellow square, and that single comparison wears three uniforms. A ratio is just a comparison of two quantities of the same kind, and this page teaches you to dress it three ways on demand.

The instruction to write the following ratio using two other notations is really asking for a costume change: keep the comparison identical, change only the symbols. Once you see the three uniforms as one ratio, the direction stops being intimidating. One worksheet line - write the following ratio using two other notations - is really three small translations wearing a single instruction.

The clearest old definition still holds. In Ray's New Higher Algebra (1866): “Ratio is the quotient which arises from dividing one quantity by another.” Compare 2 and 6: the ratio of 2 to 6 is 3, because $6 \div 2 = 3$. Ray names the two parts exactly as teachers do today: “the first is called the antecedent, and the second the consequent.” So in the ratio $a : b$, the antecedent is $a$ and the consequent is $b$.

Ray also shows the colon uniform: “the ratio of 2 to 6, is written $2 : 6$.” (Curiously, his 1866 fraction uniform put the first term in the denominator; modern books put the first term on top. When you write the following ratio using two other notations, follow the modern rule - first term first.)

Vocabulary this precise is the whole job. To write the following ratio using two other notations, you only need to know which term is the antecedent and which one is the consequent. Every era keeps the quotient; only the penmanship moves - which is why each generation can write the following ratio using two other notations without relearning the idea.

Every ratio of $a$ to $b$ has exactly three standard uniforms:

All three carry the same comparison; only the symbols change. That is why a direction like “write the following ratio using two other notations” never asks you to compute anything - it asks you to translate. Modern textbooks lean on the fraction form, because a fraction slides straight into equations and proportions later.

So most answer keys, when they write the following ratio using two other notations, put the fraction form first and the words second. A teacher who assigns this direction is really checking one thing: can you read all three uniforms without stumbling? A table like the one above is the whole toolkit for anyone about to write the following ratio using two other notations from scratch.

Take the direction “write the following ratio using two other notations” apart, and it tells you exactly what to produce.

So the whole task is a two-row translation. Given the colon form, write the following ratio using two other notations means adding the fraction form and the word form. Given the fraction form, add the colon form and the word form. Given the words, add the colon form and the fraction form. Whichever uniform you start from, to write the following ratio using two other notations is to produce the pair you do not yet have.

Problem. A park count finds 12 ducks and 7 geese. The ratio of ducks to geese is $12 : 7$. Write the following ratio using two other notations, in all three readings.

Answer. $\tfrac{12}{7}$ and “12 to 7”.

1. The colon form $12 : 7$ is given, so the two other notations are the fraction form and the word form.
2. Fraction form: the first term goes on top, so $12 : 7$ becomes $\tfrac{12}{7}$. The park asked for the raw comparison, so leave it unreduced.
3. Word form: read the colon aloud - “12 to 7”. When a park hands you a colon and asks you to write the following ratio using two other notations, reading aloud is half the answer.
4. Check the order: ducks were named first, so 12 leads in every uniform. That check is the quiet half of the direction to write the following ratio using two other notations - same numbers, same order, new symbols. Written this way, the whole switch costs two lines, not two calculations.

Problem. A class keeps its club roster in the ratio $24 : 36$, seniors to juniors. Write the following ratio using two other notations, using simplest form.

Answer. $\tfrac{2}{3}$ and “2 to 3”.

1. The direction to write the following ratio using two other notations hands you the colon form $24 : 36$, so the fraction form and the word form are wanted.
2. Simplify first: $24 : 36 = 2 : 3$, because both terms divide by 12.
3. Fraction form: $\tfrac{2}{3}$. Dividing both terms by the same number does not change the ratio, and $\tfrac{24}{36} = \tfrac{2}{3}$ confirms it.
4. Word form: “2 to 3”. Same comparison, smaller numbers - simplifying is allowed while you write the following ratio using two other notations, as long as both terms move together.
5. The direction never forbids arithmetic; it only forbids changing the comparison. $2 : 3$ and $24 : 36$ name the very same roster.

Problem. An astronaut weighs 150 pounds on Earth and 25 pounds on the Moon. The ratio of Earth weight to Moon weight is $150 : 25$. Write the following ratio using two other notations, keeping the units honest.

Astronaut numbers make the units visible - the classic setting for write the following ratio using two other notations with the units in plain sight.

Answer. $\tfrac{6}{1}$ and “6 to 1”.

1. The colon form $150 : 25$ is given; produce the fraction form and the word form.
2. Both weights carry pounds, so the pounds cancel inside the fraction: $\tfrac{150}{25} = 6$.
3. Simplest form: $\tfrac{6}{1}$ - keep it as a fraction rather than a bare 6, so the two parts of the ratio stay visible.
4. Word form: “6 to 1”. Every pound on the Moon corresponds to six pounds on Earth. Notice that to write the following ratio using two other notations here took one extra step - cancelling the matching pounds - before the translating began. Units first, translation second: the order every honest attempt at this direction should take.

A ratio is a directed comparison. Twelve ducks to seven geese is $12 : 7$; seven geese to twelve ducks is $7 : 12$ - different ratios, even though the same animals stand in the same park. The phrase “the ratio of A to B” promises that A comes first, in the colon form, on top of the fraction, and first in the words. This is the most-tested habit when you write the following ratio using two other notations - read which item is named first, and let it lead in all three uniforms.

Numbers show how badly a flip distorts. $12 : 7$ is a bit less than 2, while $7 : 12$ is a bit more than half. Flip the order and you have not restated the comparison - you have replaced it with a different one. That is why every careful answer to write the following ratio using two other notations begins with one question: which item does the sentence name first? Any drill built on write the following ratio using two other notations is quietly testing order first and translation second.

A ratio compares quantities of the same kind, and matching units behave politely: dollars over dollars cancel, pounds over pounds cancel, so $\tfrac{3}{4}$ needs no dollar signs. Mismatched units do not cancel, and the comparison becomes a rate (miles per gallon), not a ratio - a useful object, but a different one. So before you write the following ratio using two other notations, glance at the units. Same unit on both sides means let them cancel silently. Different units means convert one side first, or name the units in every uniform.

One glance, and the translation that follows cannot accidentally merge two different kinds of quantity. That is why careful attempts to write the following ratio using two other notations start by asking what kind of thing each number counts. The astronaut example only worked because both numbers counted pounds - run that same check before you write the following ratio using two other notations under any unit system.

1. Flipping the order. You see “the ratio of geese to ducks” and answer $12 : 7$ because the ducks came first in the story. Wrong - the phrase promises geese first, so $7 : 12$. Every time you write the following ratio using two other notations, re-read which item leads.

2. Putting the first term in the bottom of the fraction. Old books like Ray wrote $2 : 6$ as $\tfrac{6}{2}$, but the modern fraction form keeps the first term on top: $\tfrac{2}{6}$. If your answer has the first term in the denominator, you have silently flipped the ratio.

3. Dropping or mixing units. Treating $\tfrac{150}{25}$ as a bare 6 is fine only when the units match and cancel. Match units before simplifying, and keep a $1$ in simplified forms such as $\tfrac{6}{1}$ so the two parts stay visible.

All three mistakes share one cure: slow down at the sentence, not at the symbols. To write the following ratio using two other notations well, you translate faithfully - the arithmetic is the easy part.

Before the answer, run this four-beat routine. Beat one - find the given uniform. The direction to write the following ratio using two other notations always hands you one of the three forms; circle it.

Beat two - name the leader. Whichever quantity the sentence names first leads in every uniform you are about to write.

Beat three - write the two missing uniforms in the same order, simplifying both the same way if simplifying is wanted.

Beat four - read all three aloud. Twelve ducks to seven geese, $12 : 7$, $\tfrac{12}{7}$ - if the three readings say the same sentence, the translation is done. Beat two exists because most lost points on write the following ratio using two other notations questions are order points, not arithmetic points. Run the beats a few times and to write the following ratio using two other notations becomes a thirty-second habit rather than a guessing game. That read-back is also the cheapest way to certify any notation-switch answer before you submit it.