Lesson 5 · from §76 of the 1910 text
Products into Proportions; Alternation and Inversion
Turning a product into a proportion
If ps = qr, divide both members by qs: \dfrac pq = \dfrac rs. Dividing instead by rs gives \dfrac pr = \dfrac qs, and dividing qr = ps by pr gives \dfrac qp = \dfrac sr. So:
If the product of two numbers equals the product of two other numbers, one pair may be made the means and the other pair the extremes of a proportion.
Example. Write a^2 - 9 = 4\cdot 7 as a proportion. Factor: (a - 3)(a + 3) = 4\cdot 7. Then (a - 3) : 4 = 7 : (a + 3). Check: extremes times extremes, (a - 3)(a + 3), equals means times means, 4\cdot 7. ✓
Alternation and inversion
If \dfrac ab = \dfrac cd, then also:
- \dfrac ac = \dfrac bd, obtained by alternation (swap the means);
- \dfrac ba = \dfrac dc, obtained by inversion (turn both ratios upside down).
Both follow from ad = bc by the rule above.
Example. By alternation, 2 : 7 = 6 : 21 becomes 2 : 6 = 7 : 21. By inversion it becomes 7 : 2 = 21 : 6.
Type proportions with colons or slashes: (a - 3) : 4 = 7 : (a + 3) or (a - 3)/4 = 7/(a + 3).
Try one:
Quick check
Write 2 : 7 = 6 : 21 by alternation.
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 each as a proportion in three different ways. Type proportions like 3 : 6 = 2 : 4 or 3/6 = 2/4.
3\cdot 4 = 2\cdot 6
5\cdot 6 = 3\cdot 10
3\cdot 6 = 2\cdot x
a\cdot d = b\cdot c
(a + b)(a - b) = 2\cdot 3
(a + b)(a - b) = (a + 2)(a + 3)
Write as a proportion. Factor each side first, if necessary.
a^2 - b^2 = 2\cdot 3
a^2 - 2ab + b^2 = 3\cdot 6
a^2 - 5a + 6 = 4\cdot 2
a^2 - 7a + 12 = (a + b)^2
a^2 - 6a + 9 = a^2 - 10a + 16
x^2 - 4xy + 4y^2 = a^2 + 10ab + 25b^2
xy = 4
xy = 3
ab = 1
abc = de
mnp = xyz
ab = a + b
Write as a proportion so that x is the fourth term.
3\cdot 4 = 5\cdot x
4x = 9\cdot 7
ab = cx
px = qr
acx = bd
1 = ax
x = \dfrac{ab}{e}
xy = y + 1
Write by alternation (the first term : the third = the second : the fourth).
\tfrac23 = \tfrac46
\dfrac43 = \dfrac xy
4 : 5 = 6 : x
3 = \dfrac ab
Write by inversion (each consequent becomes the antecedent). Type subscripts as digits: P_1 is P1.
4 : 8 = 3 : 6
\dfrac32 = \dfrac ab
P : P_1 = W_1 : W
\dfrac{V_1}{V_2} = \dfrac{P_2}{P_1}
3 : \dfrac1x = \dfrac2a
\dfrac1x : \dfrac1y = \dfrac1z
\dfrac{N_1}{N_2} = \dfrac{\sqrt{T_2}}{\sqrt{T_1}}