Math SAT Level I - Chapter 4 Ratios and Proportions

In the right ΔABC, the length of leg AC is 6 inches and the length of BC is 8 inches, the ratio of AC to BC is ___ to ___, which is often written as ___ : ___ but is just the fraction ____. Like any fraction, a ratio can be reduced and can be converted to a decimal or a percent.

6 to 8, 6 : 8, 6/8 AC to BC = 6 to 8 = 6 : 8 = 6/8 AC to BC = 3 to 4 = 3 : 4 = 3/4 = 0.75 = 75%

If x > 0 and (x-3)/(9) = (8)/(x+3), then x = A. 3 B. 9 C. 27 D. 36 E. 72

B. 9 Cross multiply: (x-3)(x+3) = 72 => x² - 9 = 72 => x² = 81 => x = 9

Meri can type p pages in 1/m minutes. At this rate, how many pages can she type in m minutes? A. mp B. pm² C. 1/p D. m/p E. 1/mp

B. pm² Meri types at the rate of p pages / 1/m minutes = p/(1/m) pages per minute = mp pages per minute. Since she can type mp pages each minute, in m minutes she can type m(mp) = pm² pages.

If the ratio of the measures of two acute angles in a right triangle is 3 : 7, what is the degree measure of the smallest angle in the triangle? A. 9 B. 18 C. 27 D. 36 E. 54

C. 27 The sum of the measures of the two acute angles in a right triangle is 90°. So 3x+7x = 90 => 10x = 90 => x = 9. So the measure of the smaller acute angle is 3(9) = 27°

If a varies inversely with b, and a = 3m when b = 5n, what is b when a = 5m? A. 3/5 B. 5/3n C. 3m D. 3n E. 5n

D. 3n If a varies inversely as b, there is a constant k such that ab = k. So (3m)(5n) = k => k = 15mn. Then if a = 5m: (5m)b = k = 15mn => b = 3n

If the ratio of teachers to administrators on a committee is 3 : 5 what percent of the committee members are administrators? A. 37.5% B. 40% C. 60% D. 62.5% E. It cannot be determined from the given information

D. 62.5% Of every 8 committee members, 3 are teachers and 5 are administrators. Therefore, administrators make up 5/8 = 62.5% of the committee.

If x varies directly with y², and if x = 50 when y = 5, what is the value of x when y = 10? A. 5 B. 25 C. 50 D. 100 E. 200

E. 200 Since x varies directly with y², there is a constant k such that x/y² = k. Then k = 50/5² = 50/25 = 2 So when y = 10: 2 = x/10² = x/100 => x = 200

What is the ratio of the circumference of a circle to its radius? A. 1 B. π/2 C. √π D. π E. 2π

E. 2π By definition, π is the ratio of the circumference to the diameter of a circle, so π = C/d = C/2π = C/r = 2π

If the ratio of the diameter of circle I to the diameter of circle II is 2 : 3, what is the ratio of the area of the circle I to the area of circle II? A. 1 : 3 B. 2 : 3 C. 2 : 5 D. 3 : 5 E. 4 : 9

E. 4 : 9 If the diameters of the two circles are 2x and 3x, their radii are x and (3/2)x. So their areas are πx² and π(3/2x)² = π(9/4x²). therefore, the ratio of their areas is 1 : 9/4 = 4 : 9

Jeremy can read 36 pages per hour. At this rate, how many pages can he read in 36 minutes? A. 3.6 B. 21.6 C. 43.2 D. 72 E. 1,296

Set up a proportion: 36 pages/ 1 hour = 36 pages/ 60 minutes = x pages/ 36 minutes and cross multiply: (36)(36) = 60x => 1,296 = 60x => x = 21.6

Frank can type 600 words in 15 minutes. if Diane can type twice as fast, how many words can she type in 40 minutes?

Since Diane is twice as fast as Frank, she can type 1,200 words in 15 minutes. Now handle this rate problem exactly as you would a ratio problem. Set up a proportion and cross multiply. words/minute = 1,200/15 = x/40 => 15x = (40)(1,200) = 48,000 => x = 3,200

What is the length of EF in the figure below? [DIAGRAM A2]

Since the measures of the angles in the two triangles are the same, the triangles are similar. therefore their sides are in proportion. 10/14 = 6/EF => 10(EF) = 84 => EF = 8.4

Boyle's law states that at a fixed temperature, the volume of a gas varies inversely as the pressure on the gas. If a certain gas occupies a volume of 1.2 liters at a pressure of 20 kilograms per square meter, what volume, in liters, will the gas occupy if the pressure is increased to 30 kilograms per square meter?

Since the volume, V, varies inversely with the pressure, P, there is a constant k such that VP = k. k = (1.2)(20) = 24 => 24 = V(30) => V = 24/30 = 0.8

Assume x varies directly with y and inversely with z and that when x is 6, and y and z are each 9. What is the value of y + z when x = 9?

Since x varies directly with y, there is a constant k such that x/y = k. Then k = 6/9 => 2/3 and when x = 9, 9/y = 2/3 => 2y = 27=> y = 13.5. Since x varies inversely with z, there is a constant c such that xz = c. Then c = (6)(9) = 54. When x = 9, 9z = 54 => z = 6. The value of y + z is 13.5 + 6 = 19.5. (As x increased from 6 to 9, y increased (from 9 to 13.5) and z decreased (from 9 to 6)).

Let the measure of the four angles be 2x, 3x, 3x, and 4x. Solve for the largest angle.

Use the fact that the sum of the measures of the angles in any quadrilateral is 360° 2x + 3x +3x + 4x = 360° => 12x = 360 => x = 30. The sum of the largest angle is 4(30) = 120°

In a right triangle, the ratio of the length of the shorter leg to the length of the longer leg is 5 to 12. If the length of the hypotenuse is 65, what is the perimeter of the right triangle?

[DIAGRAM A1] Draw a right triangle and label it with the given information; then use the Pythagorean theorem. (5x)² + (12x)² = (65)² => 25x² + 144x² = 4,225 => 169x² = 4,225 => x² = 25 => x = 5. So AC = 5(5) =25, BC = 12(5) = 6, and the perimeter equals 25+60+65 = 150.

Evaluate (x+3)/(19) = (x+5)/(20)

a/b = c/d => ad = bc (x+3)/(19) = (x+5)/(20) => 20(x+3) = 19(x+5) => 20x+60 = 19x+95 => x = 35

If a apples cost c cents, find an expression that represents how many apples can be bought for d dollars.

apples/cents = a/c = x/100d 100ad = cx => x = 100ad/c

KEY FACT C1 If two numbers mare in the ratio of a : b, then for some number x, the first number is ____ and the second number is ____.

ax, bx

Solve proportions by __________. If a/b = c/d, then _________.

cross multiplying, ad = bc

Ratios can be extended to 3 or 4 or more terms. For example, we can say that the ratio of freshmen to sophomores to juniors to seniors in a school band is 3 : 4 : 5 : 4. this means that for every 3 __________ in the band there are ____ sophomores, ____ juniors, and ____ seniors.

freshmen, 4, 5, 4

A __________ is an equation that states that two ratios are equivalent. Since ratios are just fractions, any equation such as 6/8 = 3/4, in which each side is a single fraction, is a ________.

proportion

A __________ is a fraction that compares two quantities measured in different units. They often used the word "per" as in miles per hour and dollars per week.

rate

A __________ is a fraction that compares two quantities that are measured in the same units. the first quantity is the numerator, and the second quantity is the denominator.

ratio

In some problems, one variable increases as the other decreases. One example of this is inverse variation. We say that one variable varies inversely with a second variable or one variable is inversely proportional to a second variable, if their product is constant. So if y varies inversely with x, there is a constant k such that ________________.

xy = k

Rate problems are examples of direct variation. We say that one variable varies directly with a second variable, or that one variable is directly proportional to a second variable, if their quotient is a constant. So if y varies directly with x, there is a constant k such that ______________. When two quantities vary directly, as one quantity increases (or decreases), so does the other. The constant is the rate of increase or decrease.

y/x = k