Lesson 1 · from §75 of the 1910 text
Ratio
Ratio
The ratio of one number a to a second number b is the quotient of the first divided by the second, \dfrac ab. It is also written a : b, read "a is to b."
So every ratio of numbers is a fraction, and every fraction may be regarded as a ratio: \frac32, \frac{c}{2x}, \frac{a + b}{a - b}, and \frac{\sqrt2}{\sqrt[3]5} are all ratios.
The first term (the dividend, or numerator) is the antecedent; the second (the divisor, or denominator) is the consequent.
Ratios of measured quantities
Two concrete quantities have a ratio only if they're measured in the same unit. The ratio of 5 feet to 3 feet is \frac53, the unit being 1 foot. There is no ratio between 5 years and 3 feet.
Example. Simplify 2 hours : 40 minutes. Change to the same unit: 120 : 40 = \frac{120}{40} = 3.
In fact every measurement is a ratio. A distance of 100 yards is 100 times the standard yard; a sheet of 54 square inches has 54 times the surface of a square one inch on a side.
Simplifying ratios
Write the ratio as a fraction and reduce it to lowest terms.
Example. (a^2 - 4) : (a^2 + 2a) = \dfrac{(a - 2)(a + 2)}{a(a + 2)} = \dfrac{a - 2}{a}.
Example. \left(1 - \dfrac1x\right) : \left(1 - \dfrac1{x^2}\right) = \dfrac{x - 1}{x}\cdot\dfrac{x^2}{(x - 1)(x + 1)} = \dfrac{x}{x + 1}.
Try one:
Quick check
Simplify the ratio (x^2 - 1) : (x^2 + x).
From the history of algebra
Until fairly recently, nations didn't agree on standards of measurement. Among the Hebrews the unit of length was the cubit, which tradition says was the distance from the end of the king's longest finger to the point of his elbow. Our word foot is a reminder of a time when the length of the king's foot was the standard. In 1910 the standard yard of the United States was the distance between two scratches on a gun-metal bar kept at the Bureau of Standards in Washington, along with the standard pound and gallon. Since 1959 the yard has been defined as exactly 0.9144 meter.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Simplify each ratio: write it as a fraction and reduce it to lowest terms. Type 3/2, or (a - 1)/a for letters.
5 : 10
10 : 5
16a^2 : 8a
3\frac{1}{16} : 3\frac12
8\frac23 : 5\frac79
3 days : 9 hours
150 lb. : 1 ton (a ton is 2000 lb.)
(x^2 - y^2) : (x + y)
(a^3 + b^3) : (a + b)
\left(1 - \dfrac{1}{a^2}\right) : \left(1 + \dfrac1a\right)
\left(2 + \dfrac{1}{x^2}\right) : \left(4 - \dfrac{1}{x^4}\right)
\dfrac{1}{x - 3} : \dfrac{1}{x^2 - 5x + 6}
(x^2 - xy + y^2) : (x^3 + y^3)
\left(x^2 - \dfrac{y^3}{x}\right) : \left(1 - \dfrac{y}{x}\right)
\left(\dfrac{16}{x} - x\right) : \left(\dfrac{24}{x^4} + \dfrac{10}{x^3} + \dfrac{1}{x^2}\right)