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SAT II Math I : Mathematical Relationships

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Example Question #6 : How To Find Inverse Variation

varies inversely as the square of . If x = 15 , then A = 45 . Find if x = 30 (nearest tenth, if applicable).

Possible Answers:

63.6

11.3

31.8

22.5

Correct answer:

11.3

Explanation:

The variation equation is $A = \frac{K}{x^{2}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{15^{2}} = \frac{K}{225}$

$K = 45 \cdot225 = 10,125$

The equation is now $A = \frac{10,125}{x^{2}}$ .

If x = 30 , then

$A = \frac{10,125}{30^{2}} = \frac{10,125}{900} \approx 11.3$

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Example Question #7 : How To Find Inverse Variation

The current, in amperes, that a battery provides an electrical object is inversely proportional to the resistance, in ohms, of the object.

A battery provides 1.2 amperes of current to a flashlight whose resistance is measured at 20 ohms. How much current will the same battery supply to a flashlight whose resistance is measured at 16 ohms?

Possible Answers:

$2 \textrm{ A}$

$1.6 \textrm{ A}$

$0.96 \textrm{ A}$

$1.8 \textrm{ A}$

$1.5 \textrm{ A}$

Correct answer:

$1.5 \textrm{ A}$

Explanation:

If is the current and is the resistance, then we can write the variation equation for some constant of variation :

$I = \frac{k}{R}$

or, alternatively,

k=IR

To find , substitute I = 1.2,R=20 :

$k=1.2 \cdot 20 = 24$

The equation is $I = \frac{24}{R}$ . Now substitute R = 16 and solve for :

$I = \frac{24}{16} = 1.5 \textrm{ A}$

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Example Question #101 : Mathematical Relationships

The volume of a fixed mass of gas varies inversely as the atmospheric pressure, as measured in millibars, acting on it, and directly as the temperature, as measured in kelvins, acting on it.

A balloon is filled to a capacity of exactly 100 cubic meters at a time at which the temperature is 310 kelvins and the atmospheric pressure is 1,020 millibars. The balloon is released, and an hour later, the balloon is subject to a pressure of 900 millibars and a temperature of 290 kelvins. To the nearest cubic meter, what is the new volume of the balloon?

Possible Answers:

$94 \textrm{ m}^{3}$

$100 \textrm{ m}^{3}$

$83 \textrm{ m}^{3}$

$121 \textrm{ m}^{3}$

$106 \textrm{ m}^{3}$

Correct answer:

$106 \textrm{ m}^{3}$

Explanation:

If V,P,T are the volume, pressure, and temperature, then the variation equation will be, for some constant of variation ,

$V = \frac{KT}{P}$

To calculate , substitute V = 100, T = 310, P = 1,020 :

$100 = \frac{K \cdot 310}{1,020}$

$\frac{K \cdot 310}{1,020} \cdot \frac{1,020}{310} = 100 \cdot \frac{1,020}{310}$

K = 329.03

The variation equation is

$V = \frac{329.03 T}{P}$

so substitute T =290, P = 900 and solve for .

$V = \frac{329.03 \cdot 290}{900} \approx 106 \textrm{ m}^{3}$

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Example Question #8 : How To Find Inverse Variation

If is inversely proportional to and knowing that c=3 when d=5 , determine the proportionality constant.

Possible Answers:

Correct answer:

Explanation:

The general formula for inverse proportionality for this problem is

$c=\frac{k}{d}$

Given that c=3 when d=5 , we can find by plugging them into the formula.

$3=\frac{k}{5}$

Solve for by multiplying both sides by 5

$3\cdot5=\frac{k}{5}\cdot5$

15=k

So k=15 .

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Example Question #11 : How To Find Inverse Variation

The number of days needed to construct a house is inversely proportional to the number of people that help build the house. It took 28 days to build a house with 7 people. A second house is being built and it needs to be finished in 14 days. How many people are needed to make this happen?

Possible Answers:

Correct answer:

Explanation:

The general formula of inverse proportionality for this problem is

$d=\frac{k}{p}$

where is the number of days, is the proportionality constant, and is number of people.

Before finding the number of people needed to build the house in 14 days, we need to find . Given that the house can be built in 28 days with 7 people, we have

$28=\frac{k}{7}$

Multiply both sides by 7 to find .

$28\cdot7=\frac{k}{7}\cdot7$

196=k

So k=196 . Thus,

$d=\frac{196}{p}$

Now we can find the how many people are needed to build the house in 14 days.

$14=\frac{196}{p}$

Solve for . First, multiply by on both sides:

$14\cdot p=\frac{196}{p}\cdot p$

14p=196

Divide both sides by 14

$\frac{14p}{14}=\frac{196}{14}$

p=14

So it will take 14 people to complete the house in 14 days.

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Example Question #12 : How To Find Inverse Variation

The number of days to construct a house varies inversely with the number of people constructing that house. If it takes 28 days to construct a house with 6 people helping out, how long will it take if 20 people are helping out?

Possible Answers:

$12.1\ days$

$8.9\ days$

$8.4\ days$

$10\ days$

$7.5\ days$

Correct answer:

$8.4\ days$

Explanation:

The statement, 'The number of days to construct a house varies inversely with the number of people constructing that house' has the mathematical relationship $D=\frac{k}{P}$ , where D is the number of days, P is the number of people, and k is the variation constant. Given that the house can be completed in 28 days with 6 people, the k-value is calculated.

$28=\frac{k}{6}$

k=28(6)=168

This k-value can be used to find out how many days it takes to construct a house with 20 people (P = 20).

$D=\frac{168}{P}$

$D=\frac{168}{20}$

D=8.4

So it will take 8.4 days to build a house with 20 people.

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Example Question #31 : Other Mathematical Relationships

The speed of a turtle is indirectly proportional to its weight in pounds. At 10 pounds, the turtle's speed was 0.5. What is the speed of the turtle if it grew and weigh 50 pounds?

Possible Answers:

0.15

0.25

0.1

0.025

0.01

Correct answer:

0.1

Explanation:

Write the formula for the indirect proportional relationship. If one variable increases, the other variable must also decrease.

xy=k

Using speed and weight as and respectively, the equation becomes:

sw=k

Use the initial condition of the turtle's speed and weight to solve for the constant.

(0.5)(10)=k

k=5

Substitute this value back into the formula. The formula becomes:

sw=5

We want to know the speed of the turtle when it is 50 pounds. Divide the variable on both sides to isolate the speed variable.

$s=\frac{5}{w}$

Substitute the new weight of the turtle.

$s=\frac{5}{50} = \frac{1}{10} = 0.1$

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Example Question #21 : Indirect Proportionality

x varies inversely with y. When x=10, y=6. When x=3, what is y?

Possible Answers:

1.8

Correct answer:

Explanation:

Inverse variation takes the form:

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

Plugging in:

$6=\frac{k}{10}$

k=60

Then solve when x=3:

3y=60

y=20

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Example Question #22 : Indirect Proportionality

The rate of speed is indirectly proportional to the time it takes to travel somewhere. If a person walks at a rate of 3 miles per hour and it takes them 4 hours to get to their destination how far did they travel?

Possible Answers:

Correct answer:

Explanation:

For an indirect proportionality, use the equation $y=\frac{k}{x}$

If rate of speed is indirectly proportional to the time, then the speed is y (3 miles per hour) and the time is x (4 hours).

Replace those into the indirect proportionality formula

$3=\frac{k}{4}$

Solve for k by multiplying by 4 on both sides

k=12

The person traveled 12 miles.

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Example Question #41 : Other Mathematical Relationships

$3 \sqrt [3] {a} + \sqrt [3]{b} = 6$

$9 \sqrt [3] {a} +3 \sqrt [3]{b} =12$

Evaluate .

Possible Answers:

$a = \frac{27}{8}$

The system has no solution.

a = 0

$a = \frac{64}{27}$

a = 2

Correct answer:

The system has no solution.

Explanation:

Rewrite the two equations by setting $x = \sqrt [3] {a}$ and $y = \sqrt [3] {b}$ and substituting:

$3 \sqrt [3] {a} + \sqrt [3]{b} = 6$

3x+y = 6

$9 \sqrt [3] {a} +3 \sqrt [3]{b} =12$

9x+ 3y = 12

In terms of and , this is the system of linear equations:

3x+y = 6

9x+ 3y = 12

Solve the first equation for :

3x+y -3x = 6-3x

y = - 3x+ 6

Substitute this expression for in the second equation, then simplify:

9x+ 3y = 12

9x+ 3 (- 3x+ 6) = 12

9x+ 3 (- 3x) +3 ( 6) = 12

9x-9x +18 = 12

18 = 12

The statement is identically false. Therefore, the original system cannot have a solution.

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SAT II Math I : Mathematical Relationships

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #6 : How To Find Inverse Variation

Example Question #7 : How To Find Inverse Variation

Example Question #101 : Mathematical Relationships

Example Question #8 : How To Find Inverse Variation

Example Question #11 : How To Find Inverse Variation

Example Question #12 : How To Find Inverse Variation

Example Question #31 : Other Mathematical Relationships

Example Question #21 : Indirect Proportionality

Example Question #22 : Indirect Proportionality

Example Question #41 : Other Mathematical Relationships

All SAT II Math I Resources

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