2.3. Ratio, proportion, and variation

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2.3. Ratio, proportion, and variation

2.3.1. Ratio

A ratio (:) is the relation between two like numbers. For example, the ratio 5:7 can be written as a fraction, 5/7, or as division 5 ÷ 7. We read the ratio 5:7 as 5 to 7 or 5 per 7.

Definition of ratio

A ratio is a comparison of two numbers.

A ratio can be expressed by a fraction

(\frac{a}{b})

$(\frac{a}{b})$

or by a colon (:). So, the ratio of two numbers a and b can be written as a:b which is the same as the fraction

\frac{a}{b}

$\frac{a}{b}$

, where b ≠ 0.

Example 1

Reduce the ratio 20:35 to the lowest term.

Solution

20:35 = 4/7.

Example 2

Reduce the ratio 3/5:2/7 to the lowest term.

Solution

3/5:2/7 = 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10.

Example 3

Express the following statement in a ratio form: “5 out of every 17 students in the class have a new car.”

Solution

The ratio is

\frac{5}{17}

$\frac{5}{17}$

, same as 5 to 17 or 5:17.

Example 4

Find the ratio of the following:

(a) 3 to 9

(b) 7 to 13

(d) 17 to 99

Solution

(a)

\frac{3}{9} = \frac{1}{3}

$\frac{3}{9} = \frac{1}{3}$

(b)

\frac{7}{13}

$\frac{7}{13}$

(c)

\frac{11}{51}

$\frac{11}{51}$

(d)

\frac{17}{99}

$\frac{17}{99}$

Ratios can be used to find unit price or how much a service item costs.

Example 5

If 6 apples cost $3.00, find the unit price.

Solution

This is done by putting the quality on the top and the price on the bottom, then simplify, and the quality becomes 1.

$\frac{6 apples}{3 dollars} = \frac{2 apples}{1 dollar} = \frac{2 apples}{$ 1.00} = \frac{1 apple}{$ 0.50} .$ $\frac{6 apples}{3 dollars} = \frac{2 apples}{1 dollar} = \frac{2 apples}{$ 1.00} = \frac{1 apple}{$ 0.50} .$

So the unit price of these apples is 50 cents.

2.3.2. Proportion

A proportion (:: or ∝ or = ) is a statement for two ratios that are equal. It can be written in two ways:

Definition of proportion

A proportion is an equality of two ratios.

(1) 2/3 :: 4/6, that is read, 2 is to 3 as 4 is to 6,

(2) 2:3 = 4:6 or 2/3 = 4/6, that is read, 2/3 equal 4/6.

For example, the statement

\frac{1}{5} = \frac{5}{25}

$\frac{1}{5} = \frac{5}{25}$

is a proportion.

It is necessary to use the basic rules of proportions for

\frac{a}{b} = \frac{c}{d}

$\frac{a}{b} = \frac{c}{d}$

as below:

1. a × d = b × c: cross multiplication, still makes them equal

\frac{b}{a} = \frac{d}{c}

$\frac{b}{a} = \frac{d}{c}$

: by inversing of both sides, still makes them equal

\frac{a}{c} = \frac{b}{d}

$\frac{a}{c} = \frac{b}{d}$

: by replacing the denominator of the first with the numerator of the second, still makes them equal.

\frac{a + b}{b} = \frac{c + d}{d}

$\frac{a + b}{b} = \frac{c + d}{d}$

\frac{a - b}{b} = \frac{c - d}{d}

$\frac{a - b}{b} = \frac{c - d}{d}$

Example 1

Find the value of V for the following proportions:

(a)

\frac{V}{3} = \frac{2}{5}

$\frac{V}{3} = \frac{2}{5}$

(b)

\frac{5}{7} = \frac{V}{6}

$\frac{5}{7} = \frac{V}{6}$

(c)

\frac{5}{V} = \frac{8}{2}

$\frac{5}{V} = \frac{8}{2}$

(d)

\frac{2}{9} = \frac{3}{V}

$\frac{2}{9} = \frac{3}{V}$

Solution

(a)

\frac{V}{3} = \frac{2}{5}

$\frac{V}{3} = \frac{2}{5}$

(b)

\frac{5}{7} = \frac{V}{6}

$\frac{5}{7} = \frac{V}{6}$

(c)

\frac{5}{V} = \frac{8}{2}

$\frac{5}{V} = \frac{8}{2}$

(d)

\frac{2}{9} = \frac{3}{V}

$\frac{2}{9} = \frac{3}{V}$

Example 2

Solve the following proportions.

(a)

\frac{9}{5} = \frac{t + 2}{3}

$\frac{9}{5} = \frac{t + 2}{3}$

(b)

\frac{t - 1}{2} = \frac{11}{6}

$\frac{t - 1}{2} = \frac{11}{6}$

Solution

(a)

\frac{9}{5} = \frac{t + 2}{3}

$\frac{9}{5} = \frac{t + 2}{3}$

(b)

\frac{t - 1}{2} = \frac{11}{6}

$\frac{t - 1}{2} = \frac{11}{6}$

There are two kinds of proportion:

1. Direct proportion: is when two quantities are related such that when one of them increases this causes the other to increase or when one of them decreases this causes the other to decrease. For example, the greater the length, the greater the area.

2. Inverse proportion: is when two quantities are related such that when one of them increases this causes the other to decrease or when one of them decreases this causes the other to increase. For example, the greater the volume, the less the density.

2.3.3. Variation

Definition of variation

A variation is a relationship between two variables in which one is a constant multiple of the other.

Variation has two types, direct and inverse variations.

Definition of direct variation

A direct variation is a relationship between two variables in which one is a constant multiple of the other. When one variable is changed, the other changes in proportion to the first.

If y is directly proportional to x written as yαx, then y = kx, where k is called the constant of variation.

Example 1

Show that in equation y = 3x, the variable y is directly proportional to x with variation constant equal to 3.

Solution

x	y
1	3
2	6
3	9
4	12

We see that in the table every time we increase x, we note that y increases 3 times. Doubling x causes y to double, and tripling x causes y to triple.

Example 2

If y varies directly with respect to x and y = 5, when x = −3, find y when x = 12.

Solution

Let y₁ = 5, x₁ = −3, x₂ = 12, and we want to find y₂. We can use the proportion based on what is given

\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}

$\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$

. Because y varies directly with respect to x, that can lead to two equations, first, y₁ = kx₁, where k is a constant and

k = \frac{y_{2}}{x_{2}}

$k = \frac{y_{2}}{x_{2}}$

. Second, y₂ = kx₂, where

k = \frac{y_{1}}{x_{1}}

$k = \frac{y_{1}}{x_{1}}$

So substituting the values in

\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}

$\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$

can lead to the proportions

\frac{5}{- 3} = \frac{y_{2}}{12}

$\frac{5}{- 3} = \frac{y_{2}}{12}$

\frac{5}{- 3} = \frac{y_{2}}{12}

$\frac{5}{- 3} = \frac{y_{2}}{12}$

, Multiplying both side by 12

This means that when x = 12, y = −20.

Definition of joint variation

A joint variation is a relationship between one variable and a set of variables in which the one variable is directly proportional to each variable taken one at a time.

If z is directly proportional to x and y written as zαxy, then z = kxy, where k is called the constant of variation.

Example 3

Show that in equation z = 2xy, the variable z is jointly proportional to x and y with variation constant equal to 2.

Solution

x	y	z
1	1	2
2	1	4
1	2	4
2	2	8

We see that in the table every time we double x, that z will double. Also, doubling y causes z to double, and doubling both x and y causes z to increase four times as great as x or y.

Definition of inverse variation

A direct variation is a relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged.

If y is inversely proportional to x written as

y α \frac{1}{x}

$y α \frac{1}{x}$

, then

y = \frac{k}{x}

$y = \frac{k}{x}$

, where k is called constant of variation.

Example 4

Show that in equation

y = \frac{2}{x}

$y = \frac{2}{x}$

, the variable y is inversely proportional to x with variation constant equal to 2.

Solution

x	y
1	2
2	1
3	$\frac{2}{3}$ $\frac{2}{3}$
4	$\frac{1}{2}$ $\frac{1}{2}$

We see that in the table every time we increase x, y decreases with unchangeable variation constant equal to 2.

2.3. Exercises

Reduce each ratio to the lowest terms for problems 1–4.

1. 15:30

2. 6/3

3. ½::3/5

4. 3/7:4/6

For problems 5–10, find the ratio of the following.

5. 3 to 11

6. 2 to 9

7. 4 to 14

8. 5 to 15

9. 17 to 5

10. 13 to 2

For problems 11–22, find the value of V in the proportions.

11.

\frac{7}{3} = \frac{V}{2}

$\frac{7}{3} = \frac{V}{2}$

12.

\frac{1}{4} = \frac{V}{3}

$\frac{1}{4} = \frac{V}{3}$

13.

\frac{V}{2} = \frac{3}{5}

$\frac{V}{2} = \frac{3}{5}$

14.

\frac{V}{3} = \frac{6}{11}

$\frac{V}{3} = \frac{6}{11}$

15.

\frac{3}{2} = \frac{9}{V}

$\frac{3}{2} = \frac{9}{V}$

16.

\frac{13}{5} = \frac{2}{V}

$\frac{13}{5} = \frac{2}{V}$

17.

\frac{15}{V} = \frac{2}{7}

$\frac{15}{V} = \frac{2}{7}$

18.

\frac{5}{V} = \frac{9}{2}

$\frac{5}{V} = \frac{9}{2}$

19.

\frac{V + 3}{2} = \frac{1}{5}

$\frac{V + 3}{2} = \frac{1}{5}$

20.

\frac{V - 2}{5} = \frac{7}{4}

$\frac{V - 2}{5} = \frac{7}{4}$

21. V:4 = 5:13

22. V:5 = 7:3

23. Suppose 145 gallons of oil flow through a feeder pipe in 5 min. Find the flow rate in gallons per minute.

24. Suppose 180 gallons of oil flow through a feeder pipe in 8 min. Find the flow rate in gallons per minute.

25. A transformer has a voltage of 6 volts in the primary circuit and 3850 volts in the secondary circuit. Determine the ratio of the primary voltage to the secondary voltage.

26. A transformer has 65 turns in the primary coil and 640 in the secondary coil. Determine the ratio of the secondary turns to primary turns.

Identify the direct proportion and inverse proportion between the following for problems 27–30.

27. Speed and time

28. Speed and distance

29. Temperature of gas and volume

30. Volume and density

31. If

\frac{x}{y} = \frac{z}{t}

$\frac{x}{y} = \frac{z}{t}$

, x + y = 40, z = 5, and t = 4, find y. Hint: rule number 4, or by rule 1 and adding x + y both sides, then factoring.

32. If y varies directly with respect to the square of x and y = 7 when x = −1, find y when x = 6.

33. If y varies inversely with respect to x and y = 3 when x = −2, find y when x = 2.

34. If y varies inversely with respect to x and y = 5 when x = −3, find y when x = 1.

Chapter 2 Review exercises

Evaluate each expression in problems 1–4.

1. x + 4y + 3z if x = 1, y = –1, and z = 1/3.

2. −5x – 2y – z² if x = 3, y = 2, and z = 5.

\frac{x}{2} - y + \frac{t}{3} - 1

$\frac{x}{2} - y + \frac{t}{3} - 1$

if x = 2, y = 2, t = 3.

4. x³ − z₁ + k if x = –2, z₁ = 10, k = –5.

Simplify each expression in problems 5–14.

\frac{{(\frac{x}{t})}^{3}}{t^{- 2}}

$\frac{{(\frac{x}{t})}^{3}}{t^{- 2}}$

6. t³t⁸

7. (−3z⁵)(2z³)

8. (3x²y⁵z)³

9. 3(z + 2) – 7 − (8y + 1)

10. −3x²(2x³ − 1)

11.

\frac{1}{3} (12 x - 9)

$\frac{1}{3} (12 x - 9)$

12. (2xy²z³)(−3x⁻³y⁻⁷z⁻⁴)

13.

\frac{- 4 x^{3} y^{- 1}}{2 x^{11} y^{- 4}}

$\frac{- 4 x^{3} y^{- 1}}{2 x^{11} y^{- 4}}$

14.

\frac{9 x^{5} y^{11}}{3 x^{2} y^{3}}

$\frac{9 x^{5} y^{11}}{3 x^{2} y^{3}}$

In problems 15–20, classify each expression as a monomial, binomial, trinomial, or neither. Then find the degree (order) of the polynomial.

15. 4x²y⁵ − 7z⁹

16. z³x + 3xy − 4

17. z³x + 3xtz

18. xyz

19. z³ + xy + xt − 2n

20.

z^{2} - \frac{2}{t} + 1

$z^{2} - \frac{2}{t} + 1$

In problems 21 and 22, express the following expression in a single polynomial.

21. (z² − t) + (2z² − 3t)

22. (z² − t)(2z² − 3t)

In problems 23–26, perform each division.

23.

\frac{2 x^{2} z^{3}}{2 x z}

$\frac{2 x^{2} z^{3}}{2 x z}$

24.

\frac{2 x^{2} + 7 x - 1}{x + 1}

$\frac{2 x^{2} + 7 x - 1}{x + 1}$

25.

\frac{x - 2 x^{2}}{(1 - 2 x)}

$\frac{x - 2 x^{2}}{(1 - 2 x)}$

26.

\frac{x^{2} - 1}{(x - 1)}

$\frac{x^{2} - 1}{(x - 1)}$

In problems 27–30, write the following expressions in their simplest forms.

27. 7x + 11x

28. 3t – t

29. 6z – 2z + 4z + z

30. 11m – 2n + 3m – 12n

In problems 31 and 32, when t = 3, w = 1 find the numerical values of the expressions.

31. 2t − 1

32. 3w + t + 5

For problems 33–38, write the following in their simplest forms.

33. 2a × 4

34. 3x × 5y

35. x² × x⁹

36. (2x)³

37. 5t × 4t⁴

38. 3zt × 8z²t

For problems 39 and 40, find the numerical values.

39. 3x² × 4x when x = 2

40. (4tz)² − t when t = 1, z = 2

Identify the direct proportion and inverse proportion between the following for problems 41 and 42.

41. The current that flows in a circuit and the applied voltage of the circuit.

42. The current that flows in a circuit and the resistance of that circuit.

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Table of Contents for 2.3. Ratio, proportion, and variation

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2.3. Ratio, proportion, and variation

2.3.1. Ratio

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Example 5

Solution

2.3.2. Proportion

Example 1

Solution

Example 2

Solution

2.3.3. Variation

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

2.3. Exercises

Chapter 2 Review exercises

Table of Contents for
2.3. Ratio, proportion, and variation