Question 1

Find the equation of variation if the following is true.
-Suppose $p$ varies directly as $\mathrm { q }$ and $\mathrm { p } = 1$ when $\mathrm { q } = \frac { 1 } { 11 }$

Accepted Answer

Direct variation can be represented by the equation $p = kq$, where $k$ is the constant of variation. Given $p = 1$ when $q = \frac{1}{11}$, we can find $k$ by substituting these values into the equation: $1 = k \cdot \frac{1}{11}$, which gives $k = 11$. Therefore, the equation of variation is $p = 11q$.

Question 2

Solve.
-According to Ohm's law, the electric current I, in amperes, in a circuit varies directly as the voltage V. When 7 volts are applied, the current is 2 amperes. What is the current when 6 volts are applied?

Accepted Answer

Ohm's law states that I = V/R, where I is the current, V is the voltage, and R is the resistance. Given that 7 volts produce 2 amperes, we can find the resistance (R = V/I = 7/2). To find the current at 6 volts, we use the same resistance: I = V/R = 6/(7/2) = 12/7 = 1.71 amperes.

Question 3

Solve the problem.
-A machine can fill 4551 cartons of milk in 0.6 hour. How many cartons of milk can it fill per hour?

Accepted Answer

To find the number of cartons filled per hour, divide the number of cartons by the time in hours: 4551 / 0.6 = 7585 cartons per hour.

Question 4

For the pair of similar triangles, find the length of the indicated side.
-In the left triangle, the hypotenuse has length 39 , and the longer leg has length 36 . In the right triangle, the hypotenuse has length 13 , the longer leg has length 12 , and the shorter leg has length 5 . Find the length, $x$, of the shorter leg of the left triangle.

Accepted Answer

The answer of For the pair of similar triangles, find...

Question 5

Find the ratio. Simplify, if possible.
-A machine can fill 4626 boxes of cereal in 0.6 hour. How many boxes of cereal can it fill per hour?

Accepted Answer

To find the number of boxes filled per hour, divide the total boxes by the time in hours: 4626 boxes / 0.6 hour = 7710 boxes/hour.

Question 6

For the pair of similar triangles, find the length of the indicated side.
-In the outer triangle, the shortest side has length 12 , and the middle side has length 15 . In the inner triangle, the shortest side has length 4 , the middle side has length 5 , and the longest side has length 6 . Find the length, $x$, of the longest side of the outer triangle.

Accepted Answer

The answer of For the pair of similar triangles, find...

Question 7

Find the equation of variation if the following is true.
-Suppose $y$ varies directly as $x$ and $y = 0.9$ when $x = 0.1$.

Accepted Answer

Direct variation is represented by the equation $y = kx$, where $k$ is the constant of variation. Given $y = 0.9$ when $x = 0.1$, we find $k$ by dividing $y$ by $x$, resulting in $k = \frac{0.9}{0.1} = 9$. Therefore, the equation of variation is $y = 9x$.

Question 8

Solve.
-The distance D that a spring is stretched by a hanging object varies directly as the weight W of the object. If a 16-kg object stretches a spring 76 cm, find the distance when the weight is 19-kg. Round to the nearest Hundredths when necessary.

Accepted Answer

The relationship between the distance $D$ that a spring is stretched and the weight $W$ of the object is directly proportional, which can be represented as $D = k \cdot W$, where $k$ is the constant of proportionality. Given that a 16-kg object stretches the spring 76 cm, we can find $k$ by rearranging the formula: $k = \frac{D}{W} = \frac{76}{16}$. To find the distance when the weight is 19-kg, we substitute $W = 19$ and the calculated $k$ into the original formula: $D = k \cdot 19$, which gives $D = \frac{76}{16} \cdot 19$, resulting in approximately 90.25 cm.

Question 9

Solve the problem.
-In a cake recipe, the ratio of milk to flour is $\frac { 5 } { 11 }$. If 4 cups of milk are used, how many cups of flour are used?

Accepted Answer

To find the amount of flour, set up the proportion $\frac{5}{11} = \frac{4}{x}$. Solving for $x$, we get $x = \frac{44}{5}$.

Question 10

Solve the problem.
-A quality-control inspector examined 290 calculators and found 14 of them to be defective. At this rate, how many defective calculators will there be in a batch of 27,840 calculators?

Accepted Answer

The rate of defective calculators is 14 out of 290. To find the expected number of defective calculators in a batch of 27,840, use the proportion 14/290 = x/27840. Solving for x gives x = (14*27840)/290 = 1344.

Question 11

Solve the problem.
-To determine the number of fish in a lake, a park ranger catches 280 fish, tags them, and returns them to the lake. Later, 108 fish are caught, and it is found that 30 of them are tagged. Estimate the number of fish in the
Lake.

Accepted Answer

The number of fish in the lake can be estimated using the ratio of tagged to untagged fish in the second catch. If 30 out of 108 fish are tagged, and these 30 represent the 280 tagged fish released earlier, the ratio of tagged to total fish in the lake can be set up as 280/x = 30/108, where x is the total number of fish in the lake. Solving for x gives x = (280*108)/30 = 1008.

Question 12

Solve.
-The number G of gears a machine can make varies directly as the time T it operates. If it can make 5258 gears in 11 hours, how many gears can it make in 15 hours?

Accepted Answer

The relationship between the number of gears and time is direct, so $G = kT$, where $k$ is the constant of proportionality. Given $G = 5258$ when $T = 11$, we find $k = 5258 / 11 = 478$. For $T = 15$, $G = 478 \times 15 = 7170$.

Question 13

Find the equation of variation if the following is true.
-Suppose that y varies directly as z and y = 40 when z = 240.

Accepted Answer

Direct variation is given by the equation $y = k z$, where $k$ is the constant of variation. Given $y = 40$ when $z = 240$, we find $k$ by solving $40 = k \cdot 240$, which gives $k = \frac{1}{6}$. Thus, the equation of variation is $y = \frac{1}{6} z$.

Question 14

Solve the problem.
-The ratio of the weight of an object on Mars to the weight of an object on Earth is 0.4 to 1. How much will a 160-pound astronaut weigh on Mars?

Accepted Answer

To find the astronaut's weight on Mars, multiply their Earth weight by the ratio of Mars to Earth weight, which is 0.4. So, $160 \times 0.4 = 64$ pounds.

Question 15

Solve the problem.
-Maria and Charlie can deliver 100 papers in 2 hours. How long would it take them to deliver 105 papers?

Accepted Answer

Maria and Charlie deliver 100 papers in 2 hours, so their rate is 100 papers/2 hours = 50 papers/hour. To deliver 105 papers at this rate, it would take 105 papers / 50 papers/hour = 2.1 hours.

Question 16

Solve the problem.
-Sven can type 48 words per minute. How many words would he type in $\frac { 2 } { 3 }$ hour (40 minutes)?

Accepted Answer

The answer of Solve the problem.
-Sven can type 48 words...

Question 17

Find the equation of variation if the following is true.
-Suppose $m$ varies directly as $p$ and $m = 36$ when $p = 9$.

Accepted Answer

The equation of direct variation is $m = kp$, where $k$ is the constant of variation. Given $m = 36$ when $p = 9$, we find $k$ by solving $36 = k \cdot 9$, which gives $k = 4$. Therefore, the equation of variation is $m = 4p$.

Question 18

Solve the problem.
-Dr. Wong can see 8 patients in 2 hours. At this rate, how long would it take her to see 48 patients?

Accepted Answer

Dr. Wong sees 8 patients in 2 hours, so to see 48 patients (which is 6 times more), it would take 6 times longer, resulting in 12 hours.

Question 19

For the pair of similar triangles, find the length of the indicated side.
-In the upper triangle, the shortest side has length 12 , and the middle side has length 18 . In the lower triangle, the shortest side has length 8 , the middle side has length 12 , and the longest side has length 16 . Find the length, $x$, of the longest side of the upper triangle.

Accepted Answer

The answer of For the pair of similar triangles, find...

Question 20

Find the equation of variation if the following is true.
-Suppose $r$ varies directly as $\mathrm { s }$ and $\mathrm { r } = 0.0714$ when $\mathrm { s } = 1$.

Accepted Answer

Direct variation is represented by the equation $r = k \cdot s$, where $k$ is the constant of variation. Given $r = 0.0714$ when $s = 1$, we find $k = 0.0714$, making the equation $r = 0.0714s$.

Quiz 6: Rational Expressions and Equations

Find the equation of variation if the following is true. -Suppose $p$ varies directly as $\mathrm { q }$ and $\mathrm { p } = 1$ when $\mathrm { q } = \frac { 1 } { 11 }$

Solve. -According to Ohm's law, the electric current I, in amperes, in a circuit varies directly as the voltage V. When 7 volts are applied, the current is 2 amperes. What is the current when 6 volts are applied?

Solve the problem. -A machine can fill 4551 cartons of milk in 0.6 hour. How many cartons of milk can it fill per hour?

Find the ratio. Simplify, if possible. -A machine can fill 4626 boxes of cereal in 0.6 hour. How many boxes of cereal can it fill per hour?

Find the equation of variation if the following is true. -Suppose $y$ varies directly as $x$ and $y = 0.9$ when $x = 0.1$ .

Solve. -The distance D that a spring is stretched by a hanging object varies directly as the weight W of the object. If a 16-kg object stretches a spring 76 cm, find the distance when the weight is 19-kg. Round to the nearest Hundredths when necessary.

Solve the problem. -In a cake recipe, the ratio of milk to flour is $\frac { 5 } { 11 }$ . If 4 cups of milk are used, how many cups of flour are used?

Solve the problem. -A quality-control inspector examined 290 calculators and found 14 of them to be defective. At this rate, how many defective calculators will there be in a batch of 27,840 calculators?

Solve the problem. -To determine the number of fish in a lake, a park ranger catches 280 fish, tags them, and returns them to the lake. Later, 108 fish are caught, and it is found that 30 of them are tagged. Estimate the number of fish in the Lake.

Solve. -The number G of gears a machine can make varies directly as the time T it operates. If it can make 5258 gears in 11 hours, how many gears can it make in 15 hours?

Find the equation of variation if the following is true. -Suppose that y varies directly as z and y = 40 when z = 240.

Solve the problem. -The ratio of the weight of an object on Mars to the weight of an object on Earth is 0.4 to 1. How much will a 160-pound astronaut weigh on Mars?

Solve the problem. -Maria and Charlie can deliver 100 papers in 2 hours. How long would it take them to deliver 105 papers?

Solve the problem. -Sven can type 48 words per minute. How many words would he type in $\frac { 2 } { 3 }$ hour (40 minutes) ?

Find the equation of variation if the following is true. -Suppose $m$ varies directly as $p$ and $m = 36$ when $p = 9$ .

Solve the problem. -Dr. Wong can see 8 patients in 2 hours. At this rate, how long would it take her to see 48 patients?

Find the equation of variation if the following is true. -Suppose $r$ varies directly as $\mathrm { s }$ and $\mathrm { r } = 0.0714$ when $\mathrm { s } = 1$ .

Quiz 6: Rational Expressions and Equations

Find the equation of variation if the following is true. -Suppose ppp varies directly as q\mathrm { q }q and p=1\mathrm { p } = 1p=1 when q=111\mathrm { q } = \frac { 1 } { 11 }q=111​

Solve. -According to Ohm's law, the electric current I, in amperes, in a circuit varies directly as the voltage V. When 7 volts are applied, the current is 2 amperes. What is the current when 6 volts are applied?

Solve the problem. -A machine can fill 4551 cartons of milk in 0.6 hour. How many cartons of milk can it fill per hour?

Find the ratio. Simplify, if possible. -A machine can fill 4626 boxes of cereal in 0.6 hour. How many boxes of cereal can it fill per hour?

Find the equation of variation if the following is true. -Suppose yyy varies directly as xxx and y=0.9y = 0.9y=0.9 when x=0.1x = 0.1x=0.1 .

Solve. -The distance D that a spring is stretched by a hanging object varies directly as the weight W of the object. If a 16-kg object stretches a spring 76 cm, find the distance when the weight is 19-kg. Round to the nearest Hundredths when necessary.

Solve the problem. -In a cake recipe, the ratio of milk to flour is 511\frac { 5 } { 11 }115​ . If 4 cups of milk are used, how many cups of flour are used?

Solve the problem. -A quality-control inspector examined 290 calculators and found 14 of them to be defective. At this rate, how many defective calculators will there be in a batch of 27,840 calculators?

Solve the problem. -To determine the number of fish in a lake, a park ranger catches 280 fish, tags them, and returns them to the lake. Later, 108 fish are caught, and it is found that 30 of them are tagged. Estimate the number of fish in the Lake.

Solve. -The number G of gears a machine can make varies directly as the time T it operates. If it can make 5258 gears in 11 hours, how many gears can it make in 15 hours?

Find the equation of variation if the following is true. -Suppose that y varies directly as z and y = 40 when z = 240.

Solve the problem. -The ratio of the weight of an object on Mars to the weight of an object on Earth is 0.4 to 1. How much will a 160-pound astronaut weigh on Mars?

Solve the problem. -Maria and Charlie can deliver 100 papers in 2 hours. How long would it take them to deliver 105 papers?

Solve the problem. -Sven can type 48 words per minute. How many words would he type in 23\frac { 2 } { 3 }32​ hour (40 minutes) ?

Find the equation of variation if the following is true. -Suppose mmm varies directly as ppp and m=36m = 36m=36 when p=9p = 9p=9 .

Solve the problem. -Dr. Wong can see 8 patients in 2 hours. At this rate, how long would it take her to see 48 patients?

Find the equation of variation if the following is true. -Suppose rrr varies directly as s\mathrm { s }s and r=0.0714\mathrm { r } = 0.0714r=0.0714 when s=1\mathrm { s } = 1s=1 .

Find the equation of variation if the following is true. -Suppose $p$ varies directly as $\mathrm { q }$ and $\mathrm { p } = 1$ when $\mathrm { q } = \frac { 1 } { 11 }$

Find the equation of variation if the following is true. -Suppose $y$ varies directly as $x$ and $y = 0.9$ when $x = 0.1$ .

Solve the problem. -In a cake recipe, the ratio of milk to flour is $\frac { 5 } { 11 }$ . If 4 cups of milk are used, how many cups of flour are used?

Solve the problem. -Sven can type 48 words per minute. How many words would he type in $\frac { 2 } { 3 }$ hour (40 minutes) ?

Find the equation of variation if the following is true. -Suppose $m$ varies directly as $p$ and $m = 36$ when $p = 9$ .

Find the equation of variation if the following is true. -Suppose $r$ varies directly as $\mathrm { s }$ and $\mathrm { r } = 0.0714$ when $\mathrm { s } = 1$ .