Other Mathematical Relationships - SAT Subject Test in Math I

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Question

The quantity x varies directly with y. If x is 26 when y is 100, find x when y is 200.

Answer

We must set up a proportion. Since x varies directly with y, when y is multiplied by 2, x is also multiplied by 2. 26 times 2 is 52.

Direct variation: $\frac{x_1}{x_2}=\frac{y_1}{y_2}$

$\frac{26}{x_2}=\frac{100}{200}$

$\frac{26}{x_2}=\frac{1}{2}$

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Question

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?

Answer

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.

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

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

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

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

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

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Question

varies directly with , and inversely with the square root of .

If and , then .

Find if and .

Answer

The variation equation can be written as below. Direct variation will put in the numerator, while inverse variation will put $\sqrt{z}$ in the denominator. is the constant that defines the variation.

$x=k* \frac{y}{\sqrt{z}}$

To find constant of variation, , substitute the values from the first scenario given in the question.

$10=k* \frac{6}{\sqrt{64}}=k* \frac{6}{8}=k* \frac{3}{4}$

$k=10\div\frac{3}{4}}= \frac{40}{3}$

We can plug this value into our variation equation.

$x=\frac{40}{3}\cdot \frac{y}{\sqrt{z}}$

Now we can solve for given the values in the second scenario of the question.

$x=\frac{40}{3}* \frac{9}{\sqrt{144}}=\frac{40}{3}* \frac{9}{12}=10$

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Question

Sunshine paint is made by mixing three parts yellow paint and one part red paint. How many gallons of yellow paint should be mixed with two quarts of red paint?

(1 gallon = 4 quarts)

Answer

First set up the proportion:

$\frac{3; parts ; yellow}{1 ; part; red}=\frac{x ; quarts ; yellow}{2 ; quarts ; red}$

x =

Then convert this to gallons:

$6 ; quarts; \cdot \frac{1 ; gallon}{4; quarts}= 1.50; gallons$

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Question

Sarah notices her map has a scale of $\frac{1}{4}; in=1; mile$ . She measures between Beaver Falls and Chipmonk Cove. How far apart are the cities?

Answer

$\frac{1}{4}; in=1; mile$ is the same as

So to find out the distance between the cities

$12.5; in \cdot \frac{4; miles}{1;in }=50; miles$

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Question

varies inversely as the square root of . If , then . Find if (nearest tenth, if applicable).

Answer

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

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

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Question

varies inversely as the square of . If , then . Find if (nearest tenth, if applicable).

Answer

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 , then

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

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Question

If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?

Answer

Let be the mass of the weight and the elongation of the spring. Then for some constant of variation ,

We can find by setting from the first situation:

$7.2 = k \cdot 20$

$k = 7.2 \div 20 = 0.36$

so

In the second situation, we set and solve for :

$L = 0.36 \cdot32 =11.52$

which rounds to 11.5 centimeters.

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Question

If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?

Answer

Let be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation ,

.

We can find by setting :

$6.6 = k \cdot 33$

$k = 6.6 \div 33 = 0.2$

Therefore .

Set and solve for :

$M = 10 \div 0.2 = 50$ kilograms

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Question

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?

Answer

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

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

To calculate , substitute :

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

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

The variation equation is

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

so substitute and solve for .

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

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Question

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?

Answer

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,

To find , substitute :

$k=1.2 \cdot 20 = 24$

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

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

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Question

If is directly proportional to and when at , what is the value of the constant of proportionality?

Answer

The general formula for direct proportionality is

where is the proportionality constant. To find the value of this , we plug in and

Solve for by dividing both sides by 12

$\frac{24}{12}=\frac{12k}{12}$

So .

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Question

The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn $48.

Answer

The general formula for direct proportionality is

where is how much money you earned, is the proportionality constant, and is the number of hours worked.

Before we can figure out how many hours you need to work to earn $48, we need to find the value of . It is given that you earned $32 by working 4 hours. Plug these values into the formula

Solve for by dividing both sides by 4.

$\frac{32}{4}=\frac{4k}{4}$

So . We can use this to find out the hours you need to work to earn $48. With , we have

Plug in $48.

Divide both sides by 8

$\frac{48}{8}=\frac{8h}{8}$

So you will need to work 6 hours to earn $48.

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Question

If is inversely proportional to and knowing that when , determine the proportionality constant.

Answer

The general formula for inverse proportionality for this problem is

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

Given that when , we can find by plugging them into the formula.

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

Solve for by multiplying both sides by 5

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

So .

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Question

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?

Answer

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 .

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

So . 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:

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

Divide both sides by 14

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

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

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Question

Sally currently has 192 books. Three months ago, she had 160 books. By what percentage did her book collection increase over the past three months?

Answer

To find the percentage increase, divide the number of new books by the original amount of books:

She has 32 additional new books; she originally had 160.

$\frac{32}{160} = \frac{16}{80} = 0.20=20%$

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Question

Multiply in modulo 6:

$4 \times 5 \times 3$

Answer

In modulo 6 arithmetic, a number is congruent to the reainder of its division by 6.

Therefore, since $4 \times 5 \times 3 = 60$ and $60 \div 6 = 10 \textrm{ R }0$ ,

$4 \times 5 \times 3 \equiv 0 \mod 6$ .

The correct response is 0.

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Question

Which is an example of a set that is not closed under addition?

Answer

A set is closed under addition if and only if the sum of any two (not necessarily distinct) elements of the set is also an element of the set.

$\left { 0\right }$ is closed under addition, since

The set of all negative integers is closed under addition, since any two negative integers can be added to yield a third negative integer.

The set of all positive even integers is closed under addition, since any two positive even integers can be added to yield a third positive even integer.

The remaining set is the set of all integers between 1 and 10 inclusive. It is not closed under addition, as can be seen by this counterexample:

$1,10 \in \left {1,2,3...10 \right }$

but

$1+10 = 11 otin \left {1,2,3...10 \right }$

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Question

varies directly as the square root of .

If then . To the nearest tenth, calculate if .

Answer

varies directly as $\sqrt{L}$ , which means that for some constant of variation ,

$\frac{T}{\sqrt{L}} = k$

We can write this relationship alternatively as

$\frac{T_{1}}{\sqrt{L_{1}}} = \frac{T_{2}}{\sqrt{L_{2}}}$

where the initial conditions can be substituted on the left side and final conditions, on the right. We will be solving for $T_{2}$ in the equation

$\frac{13}{\sqrt{42}} = \frac{T_{2}}{\sqrt{12}}$

$\frac{13}{\sqrt{42}} \cdot \sqrt{12}= \frac{T_{2}}{\sqrt{12}} \cdot \sqrt{12}$

$T_{2} = \frac{13\sqrt{12}}{\sqrt{42}} \approx \frac{13 \cdot 3.4641}{6.4807} \approx 6.9$

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Question

varies inversely as the square of and directly as the cube of .

If and , then . Calculate if .

Answer

varies inversely as $m^{2}$ and directly as the cube of $b^{3}$ . This means that for some constant of variation ,

$\frac{Nm^{2}}{b^{3}} = k$

We can write this relationship alternatively as

$\frac{N_{1}m_{1}^{2}}{b_{1}^{3}} = \frac{N_{2}m_{2}^{2}}{b_{2}^{3}}$

where the initial conditions can be substituted on the left side and final conditions, on the right. We will be solving for $N_{2}$ in the equation

$\frac{9,000 \cdot 6^{2}}{2^{3}} = \frac{N_{2} \cdot 4^{2}}{4^{3}}$

$\frac{9,000 \cdot 36}{8} = \frac{N_{2} \cdot 16}{64}$

$40,500 = \frac{N_{2} }{4}$

$N_{2} = 40,500 \cdot 4 = 162,000$

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