Practice Makes Perfect Algebra - Carolyn Wheater [24]
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.
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Solving proportions
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. You can cross multiply to create an equation that you can solve for the missing term of a proportion.
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EXERCISE 10.2
Solve each proportion to find the value of the variable.
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Variation
Variation looks at how quantities change, specifically in relation to one another. There are two basic variation relationships, direct variation and inverse variation, which can be combined 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. 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.
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.
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EXERCISE 10.3
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?
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Inverse variation
Quantities that vary inversely