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SAT II Math I : Other Mathematical Relationships

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Example Question #31 :Basic Single Variable Algebra

$\small x$ varies directly with $\small y$ and $\small z^{2}$ . If $\small y = \frac{1}{3}$ and $\small z = 3$ , then $\small x = 3$ . Find $\small x$ if $\small y = 12$ and $\small z = 2$ .

Possible Answers:

None of these answers are correct

Correct answer:

Explanation:

From the relationship of $\small x$ , $\small y$ , and $\small z$ ; the equation of variation can be written as:

$\small x = k \cdot y \cdot z^{2}$

Using the values given in the first scenario, we can solve for k:

$\small 3 = k \cdot \frac{1}{3} \cdot 3^{2} = k \cdot \frac{1}{3} \cdot 9 = k \cdot 3$

$\small k = 1$

Using the value of k and the values of y and z, we can solve for x:

$\small x = 1 \cdot 12 \cdot 2^{2} = 12 \cdot 4 = 48$

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Example Question #32 :Basic Single Variable Algebra

varies inversely withand the square root of. When x=4 and z=81 , $y=\frac{2}{9}$ . Findwhen x=8 and z=4 .

Possible Answers:

None of these answers are correct

$\small \frac{1}{4}$

$\small \frac{1}{2}$

Correct answer:

$\small \frac{1}{2}$

Explanation:

First, we can create an equation of variation from the the relationships given:

$y = k \cdot \frac{1}{x \cdot \sqrt{z}}$

Next, we substitute the values given in the first scenario to solve for $\small k$ :

$\frac{2}{9} = k \cdot \frac{1}{4 \cdot \sqrt{81}} = k \cdot \frac{1}{4 \cdot 9} = k \cdot \frac{1}{36}$

$\small \rightarrow k = 36 \cdot \frac{2}{9} = \frac{36}{9} \cdot 2 = 4 \cdot 2 = 8$

Using the value for, we can now use the second values forandto solve for:

$\small Y = 8 \cdot \frac{1}{8 \cdot \sqrt{4}} = 8 \cdot \frac{1}{8 \cdot 2} = 8 \cdot \frac{1}{16} = \frac{1}{2}$

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Example Question #33 :Basic Single Variable Algebra

varies directly with $z^{2}$ and the square root of. If y=100 , and $\small z=12$ then x=5 . Find the value ofif y=4 and z=6 .

Possible Answers:

$\small \frac{1}{2}$

$\small \frac{16 \sqrt6}{288}$

None of these answers are correct

$\small \frac{1}{4}$

Correct answer:

$\small \frac{1}{4}$

Explanation:

From the relationship given, we can set up the variation equation

$\small x = k \cdot \sqrt{y} \cdot z^{2}$

Using the first relationship, we can then solve for $\small k$

$\small 5 = k \cdot \sqrt{100} \cdot 12^{2}$

$\small 5 = k \cdot 10 \cdot 144$

$\small k = \frac{5}{10 \cdot 144} = \frac{1}{2\cdot144} = \frac{1}{288}$

Now using the values from the second relationship, we can solve for x

$\small x = \frac{1}{288}\cdot\sqrt4\cdot6^2 = \frac{1}{288}\cdot2\cdot36 = \frac{72}{288} = \frac{1}{4}$

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Example Question #6 :How To Find Inverse Variation

varies inversely as the square of. If x = 15 , then A = 45 . Findif 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:

Ifis the current andis 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 #2 :How To Find Direct Variation

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:

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

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

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

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

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

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

Ifis inversely proportional toand 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 findby plugging them into the formula.

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

Solve forby 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

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

whereis the number of days,is the proportionality constant, andis 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 byon 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 asandrespectively, the equation becomes:

sw=k

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

(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 variableon 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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SAT II Math I : Other Mathematical Relationships

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

All SAT II Math I Resources

Example Questions

Example Question #31 :Basic Single Variable Algebra

Example Question #32 :Basic Single Variable Algebra

Example Question #33 :Basic Single Variable Algebra

Example Question #6 :How To Find Inverse Variation

Example Question #7 :How To Find Inverse Variation

Example Question #2 :How To Find Direct Variation

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

All SAT II Math I Resources

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