Lesson 6 · from §76 of the 1910 text
Composition and Division
Composition and division
If a, b, c, and d are in proportion, they are also in proportion by addition, by subtraction, and by addition and subtraction.
Addition. From \dfrac ab = \dfrac cd, add 1 to both members: \dfrac ab + 1 = \dfrac cd + 1, or \frac{a + b}{b} = \frac{c + d}{d}.
Subtraction. Subtract 1 from both members: \frac{a - b}{b} = \frac{c - d}{d}.
Addition and subtraction. Divide the first result by the second: \frac{a + b}{a - b} = \frac{c + d}{c - d}.
These three are often called composition, division, and composition and division.
Example. By addition, 3 : 5 = 9 : 15 gives 8 : 5 = 24 : 15; by subtraction, -2 : 5 = -6 : 15; by addition and subtraction, 8 : (-2) = 24 : (-6).
A shortcut for solving
Composition and division can make an equation easy. Solve \dfrac{x + 3}{x - 3} = \dfrac{7}{5}. By addition and subtraction, \frac{(x + 3) + (x - 3)}{(x + 3) - (x - 3)} = \frac{7 + 5}{7 - 5}, \quad \frac{2x}{6} = \frac{12}{2}, \quad x = 18.
Proofs
To prove a new proportion from \frac ab = \frac cd, look for a proportion you can already get (by inversion, alternation, or multiplying both members by a number) that turns into the new one when you add or subtract 1.
Try one:
Quick check
\dfrac{x + 3}{x - 3} = \dfrac75. Solve for x (by addition and subtraction, or any way you like).
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.
Write by addition: \dfrac ab = \dfrac cd becomes \dfrac{a + b}{b} = \dfrac{c + d}{d}. You may leave sums unworked, like (2 + 3)/3 = (4 + 6)/6.
\tfrac23 = \tfrac46
4 : 12 = 8 : 24
a : x = 1 : 2
4 : 3 = f : x
\dfrac{A_1}{A_2} = \dfrac{S_1^2}{S_2^2}
Write Exercises 1–4, preceding, by subtraction.
Write Exercises 1–4, preceding, by addition and subtraction.
Proofs. Work out each proof on paper; then type the proportion asked for.
If \dfrac ab = \dfrac cd, prove \dfrac{a + b}{a} = \dfrac{c + d}{c}. Adding 1 to both members of which proportion (obtained from \dfrac ab = \dfrac cd) gives this result?
If \dfrac ab = \dfrac cd, prove \dfrac{a - b}{a} = \dfrac{c - d}{c}. Subtracting both members of which proportion (obtained from \dfrac ab = \dfrac cd) from 1 gives this result?
If \dfrac ab = \dfrac cd, prove \dfrac{2a + b}{b} = \dfrac{2c + d}{d}. Adding 1 to both members of which proportion (obtained from \dfrac ab = \dfrac cd by multiplying both members by a number) gives this result?
If \dfrac ab = \dfrac cd, prove \dfrac{a + 3b}{b} = \dfrac{c + 3d}{d}. Which proportion, obtained from \dfrac ab = \dfrac cd by dividing both members by a number, gives this result after adding 1 to both members and then multiplying both members by 3?
Write each by addition and subtraction, and solve the resulting equation for x.
\dfrac{3x + 4}{3x - 4} = \dfrac{3 + 2}{3 - 2}
\dfrac{6x + 3}{6x - 3} = \dfrac{2a + b}{2a - b}