# Ratio

In algebra, a ratio is the relationship between two quantities. It is expressed as the quotient of two numbers, or as two numbers separated by a colon (pronounced "to"). A number that can be written as a ratio of two integers is a rational number.

Examples:

• If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
• In a benzene ring, atoms of carbon and hydrogen exist in a one-to-one ratio with each other; there are the same number of each. The carbon:hydrogen ratio in naphthalene is 5:4. The ratio of oxygen atoms to hydrogen atoms in water is 1:2.
• Jupiter is 318 times the size of Earth.
• The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.

Note the use of words such as "times", "parts", "number", etc. This occurs because ratios are unitless; the units cancel out of the ratio. e.g. 3 kg / 5 kg = 3000 g / 5000 g = 3/5.

Ratios are not exactly the same thing as vulgar fractions. For example, if I have three pennies and five nickels, then the ratio of pennies to nickels is 3:5 or 3/5. This indicates that there are three fifths as many pennies as nickels. However the fraction of coins which are pennies is 3/(3+5) = 3/8. Thus the chances of a randomly selected coin being a penny are three in eight.

The most common thing to do with ratios is multiply them. For example, there are 60 minutes in an hour and 24 hours in a day. Therefore, (since 60 min/hr × 24 hr/day = 1440 min/day) there are 1440 minutes in every day. Note that the intermediate unit, hours, canceled out of the expression. Also, each fraction in the expression was equal to one, which is how we know that their product (1440 min/day) is also equal to one. Similarly, if we want to know the number of minutes in a week, then we can multiply 7 days by 1440 minutes per day (1440 min/day = one, the multiplicative identity) to arrive at an answer of 10080 minutes.

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