Proportion and variation
Mathematics is more than arithmetic, as you certainly have seen. One of the things math often does is compare quantities, and those comparisons can be based on different operations. The statement that x is 4 more than y is based on adding or subtracting, but saying a is 3 times the size of b is based on multiplication or division. A ratio is a comparison of two numbers by division. The relationship between 10 and 5 or between 26 and 13 can be expressed as a ratio: 10 : 5 or 
Both of these are equal to 2 : 1.
Using ratios and extended ratios
If two quantities are in the ratio a : b, it’s not assured that they are exactly equal to a and b, but they are multiples of a and b. As a result, you can represent them as ax and bx and use those expressions to write an equation about the numbers. If two numbers add to 50 and are in ratio 3 : 7, you can represent the numbers as 3x and 7x and write 3x + 7x = 50. You’ll find that x = 5, so the actual numbers are 3x = 3 ⋅ 5 and 7x = 7 ⋅ 5, or 15 and 35.
An extended ratio is a comparison of three or more numbers, usually written in the form a : b : c. If the measurements of the angles of a triangle are in ratio 2 : 3 : 5, you can represent the measures of the angles by 2x, 3x, and 5x and add 2x + 3x + 5x = 180°. Once you solve and find x = 18°, remember to multiply by the appropriate coefficients to find the angle measures: 2x = 36°, 3x = 54°, and 5x = 90°.
Use ratios to solve each problem.
1. Find the number of degrees in each angle of a triangle if the angles are in the ratio 3 : 4 : 5.
2. A piece of wood 20 ft long needs to be cut into two pieces that are in ratio 2 : 3. How long should each piece be?
3. Two numbers are in ratio 8 : 3 and their difference is 65. Find the numbers.
4. Laura determined that the perfect recipe for her raspberry limeade was to mix raspberry juice and lime juice in a 5 : 7 ratio. How much raspberry juice will she need to make 48 oz of the mixture?
5. Three numbers are in ratio 3 : 4 : 8. The sum of the two larger numbers exceeds twice the smallest by 48. Find the numbers.
6. Two numbers are in the ratio 5 : 6. If 8 is added to each of the numbers, they will be in the ratio 7 : 8. Find the numbers.
7. Two numbers are in the ratio 3 : 7. If 1 is added to the smaller number and 7 is added to the larger, they will be in the ratio 1 : 3. Find the numbers.
8. What should be added to both 9 and 29 to produce numbers that are in the ratio 3 : 4?
9. The numerator and denominator of a fraction are in the ratio 2 : 5. If 2 is subtracted from both the numerator and denominator, the resulting fraction is equal to 
Find the original numerator and denominator.
10. The larger of two numbers is 2 more than 3 times the smaller. If 3 is added to the smaller number and 1 is added to the larger, they will then be in the ratio 3 : 7. Find the numbers.
Solving proportions
A proportion is a statement that two ratios are equal, or an equation of the form 
Any two equal ratios form a proportion. In a proportion like 10 : 5 = 2 : 1, the numbers on the ends, 10 and 1, are called the extremes, and the numbers in the middle, 5 and 2, are called the means. When the ratios are written as fractions, the proportion is 
In any proportion, the product of the means is equal to the product of the extremes. For example, in 10 : 5 = 2 : 1, 10 × 1 = 5 × 2 and in 
3 × 28 = 7 × 12 = 84. When one term of the proportion is unknown, you can cross-multiply to create an equation that you can solve for the missing term of a proportion.
Solve each proportion to find the value of the variable.
1. 
3. 
2. 
4. 
5. 
8. 
6. 
9. 
7. 
10. 
Variation
Variation looks at how quantities change, specifically in relation to one another. When you observe two related variables, does one increase as the other increases? Do they increase at the same rate, or different rates? How do you describe that relationship? Or does one get larger as the other gets smaller? How do you describe that activity with an equation? There are two basic variation relationships, direct variation, when both increase (or both decrease), and inverse variation, when the quantities move in opposite directions. There are more complex relationships, of course, but those can be built by combining direct and inverse variation in different ways.
Direct variation
When two quantities vary directly, they increase or decrease together. If 2 hamburgers cost $3 and 4 hamburgers cost $6, the total cost of the hamburgers varies directly with the number of hamburgers you buy. The number of hamburgers goes up, and the total cost goes up. But the number of hamburgers increased by 2 hamburgers, and the total cost increased by $3. There has to be a number in the equation that talks about how much things change. If y varies directly as x, there is a constant k such that y = kx, or 
This constant of variation, as it’s called, is the ratio of a y-value to its corresponding x-value. In the hamburger example, 
so k = 1.5. You can say total cost = 1.5 × number of burgers.
If you know that two quantities are directly related, you can plug in known values of x and y to find k, and once you know k, you can apply the relationship to other values of x or y. For example, if y varies directly as x, and y = 12 when x = 2, you can find that k = 6, either by solving 12 = k · 2 or by dividing 
Once you know that k = 6, if you’re told that x has changed to 5, you can determine that y = 6 · 5 = 30. If you find that y has changed to 42, you can solve 42 = 6x and see that x = 7.
Use the direct variation equation y = kx to find k, and then find the value of the variable requested.
1. If y varies directly as x and y = 12 when x = 4, find y when x = 14.
2. If y varies directly as x and y = 5 when x = 20, find x when y = 25.
3. If t varies directly as r and t = 52 when r = 13, find t when r = 78.
4. If a varies directly as b and a = 17 when b = 51, find b when a = 425.
5. If y varies directly as the square of x and y = 28 when x = 2, find y when x = 10.
6. The voltage V in an electric circuit varies directly with the current I when I = 40 A, V = 0.06 V. Find V when I = 6 A.
7. The distance covered in a fixed time varies directly with the speed of travel. If you can travel 117 mi at 45 mph, how far will you travel in the same time if you increase your speed to 55 mph?
8. The time that passes between the moment a flash of lightning is seen and the moment a clap of thunder is heard varies directly with the observer’s distance from the center of the storm. If 10 s elapse between the lightning and the thunder, the storm is 2 mi away. How far is the storm if 3 s pass between the flash and the sound of thunder?
9. The volume of a gas under a constant pressure varies directly with its temperature. At 18°C, a gas has a volume of 152 cm3. What is the volume when the temperature is 36°C?
10. If a car uses 7 gal of gas to travel 119 mi at a certain speed, how far can it travel on 10 gal of gas if it travels at the same speed?
Inverse variation
Quantities that vary inversely move in opposite directions. When one quantity increases, the other decreases. If y varies inversely as x, then there is a constant k such that 
or xy = k. You can use known values to find k and then calculate x or y, just as you did for direct variation, if you’re given the value of the other variable. If y varies directly with x, and y = 9 when x = 2, substituting those values into xy = k tells you that k = 18. If x increases to 6, 
Use the inverse variation equation 
to find k, and then find the value of the variable requested.
1. If y varies inversely with x and y = 4 when x = 6, find y when x = 8.
2. If y varies inversely with x and y = 8 when x = 4, find x when y = 2.
3. If y varies inversely with x and y = 24 when x = 3, find y when x = 9.
4. If r varies inversely with t and t = 11 when r = 12, find t when r = 3.
5. If a varies inversely with b and a = 54 when b = 10, find a when b = 45.
6. Given that v varies inversely with t, and v = 32 when t = 4, what is the value of v when t = 24?
7. If w varies inversely with l and w = 22 when l = 4, what is the value of w when l = 22.
8. If N varies inversely with B and N = 3 when B = 8, find the value of B for which N = 144.
9. If y varies inversely with x and y = 39 when x = 6, find the value of x that produces a y-value of 18.
10. If a = 18 and b = 16, and a and b vary inversely, what is the value of a when b is 12?
Joint and combined variation
A quantity varies jointly with two or more other quantities if it varies directly with their product; that is, y varies jointly with x and z if there is a constant k such that y = kxz. If x is held constant, y varies directly with z. If z is held constant, y varies directly with x.
Combined variation occurs when y varies directly with x and inversely with z. The equation for combined variation has the form 
If you hold x constant, y will vary inversely with z. If you hold z constant, y will vary directly with x.
In both joint and combined variation, you can find the value of k just as you did with direct and inverse variation. Choose the correct equation, plug in the known values, and solve for k. Then, just as before, if one or more variables change, plug in the new values along with k and solve for whatever is missing.
Use the appropriate variation equation to find k, and then find the value of the variable requested.
1. If y varies jointly with x and z, and y = 84 when x = 7 and z = 3, find y when x = 9 and z = 5.
2. If y varies jointly with x and z, and y = 165 when x = 6 and z = 11, find x when y = 200 and z = 10.
3. If y varies jointly with x and z, and y = 3375 when x = 3 and z = 9, find z when x = 12 and y = 10,500.
4. If y varies directly with x and inversely with z, and y = 9 when x = 270 and z = 12, find y when x = 100 and z = 8.
5. If y varies directly with x and inversely with z, and y = 53 when x = 3 and z = 12, find x when y = 159 and z = 28.
6. If y varies directly with x and inversely with z, and y = 4 when x = 8 and z = 14, find z when x = 36 and y = 6.
7. The volume of a solid varies jointly with the area of its base and its altitude. When the area of the base is 12 cm2 and the altitude is 16 cm, the volume is 64 cm3. Find the altitude when the volume is 60 cm3 and the area of the base is 9 cm2.
8. The gravitational force between two bodies varies directly with the masses of the bodies and inversely with the square of the distance between them. When a 70-kg person stands on the surface of the earth, 6378 km from the center of the earth, the force of gravity is 686 N. Find the force of gravity acting on the same person when the person is in a plane 40,000 ft (about 12 km) above the surface of the earth.
9. The volume of a gas varies directly with temperature and inversely with pressure. At a temperature of 450°F and a pressure of 40 psi, a gas has a volume of 10 ft3. What is its volume if temperature is reduced to 440°F and pressure is raised to 50 psi?
10. The electric resistance of a wire varies directly with its length and inversely with the square of its diameter. A wire 80 ft long with a diameter of 
in. has a resistance of 
ohm. What is the resistance in a piece of the same type of wire that is 120 ft long and has a diameter of 
in.?