Dependent and independent variables

Learning goals

Identifies independent and dependent variables in a relationship.
Explains how the dependent variable changes in response to the independent variable.
Keeps each variable linked to its quantity and measurement unit.
Connects the variables to tables, graphs, or equations.

Difficulty: 1 – 5

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Dependent and independent variables 1

A faucet fills a tub over time. Let $m$ be the number of minutes and $g$ be the gallons added. Which variable is independent, and which is dependent?

Hints

- Ask which quantity can be selected first. - Identify which quantity changes as the other one changes.

Solution

1. The number of minutes $m$ is the input that can be chosen, so it is independent. 2. The gallons added $g$ change in response to time, so they are dependent.

Answer

Independent variable: $m$ Dependent variable: $g$

Dependent and independent variables 2

In the ordered pair $(3, 15)$ for the relationship $c=5n$, $n$ is the number of notebooks and $c$ is the cost in dollars. Explain what each coordinate means and identify the independent variable.

Hints

- Match the coordinate order to the variable order in the relationship. - Use the context to decide which quantity determines the other.

Solution

1. The first coordinate gives $n=3$ notebooks. 2. The second coordinate gives $c=15$ dollars. 3. The notebook count $n$ is independent, and the cost $c$ is dependent.

Answer

The pair means $3$ notebooks cost $\$15$. The independent variable is $n$, and the dependent variable is $c$.

Dependent and independent variables 3

For a cube, $V=s^3$, where $s$ is side length and $V$ is volume. Explain why $V$ is the dependent variable.

Hints

- Ask which value must be known before the other can be calculated. - Identify which quantity is produced by the formula.

Solution

1. A value of $s$ determines one specific value of $V$ through $V=s^3$. 2. Therefore, $V$ depends on the chosen side length $s$.

Answer

$V$ is dependent because its value is determined by the chosen side length $s$.

Dependent and independent variables 4

Each sticker sheet has $5$ stickers. Let $s$ be the number of sheets and $T$ the total number of stickers. Which variable is independent? Which is dependent? Explain.

Hints

- Which amount can someone choose? - Which amount must change as a result of that choice?

Solution

1. The number of sheets can be chosen first, so $s$ is independent. 2. The sticker total follows from $T=5s$, so $T$ is dependent.

Answer

Independent: $s$, the number of sheets. Dependent: $T$, the total number of stickers, because $T$ depends on $s$.

Dependent and independent variables 5

A bike rental shop charges $\$3$ for each hour. The rental time is $h$ hours and the cost is $C$ dollars. Which quantity depends on the other? What happens to $C$ when $h$ increases by one hour?

Hints

- Which quantity is measured first? - What does “per hour” tell you about changing the cost?

Solution

1. The charge is based on rental time, so $h$ is the independent quantity and $C$ depends on it. 2. Each additional hour increases the cost by $\$3$.

Answer

$C$ depends on $h$. Each extra hour increases $C$ by $\$3$.

Dependent and independent variables 6

A garden has $4$ plants in each row. Let $r$ be the number of rows and $p$ the total plants. Leo says, “$p$ is independent because it is bigger than $r$.” Is he correct? Explain.

Hints

- Does the larger number always have to be the independent one? - Decide which choice determines how many plants are needed.

Solution

1. Independent and dependent describe which quantity determines the other, not which value is larger. 2. Choosing $r$ rows determines $p=4r$, so $r$ is independent and $p$ is dependent.

Answer

No. $r$ is independent; $p$ depends on it. Their sizes do not determine their roles.

Dependent and independent variables 7

Lucas builds staircases from cubes. Each new step adds a row with two more cubes than the row above it. He records his results. <table> <tr><th>Staircase height</th><th>1</th><th>2</th><th>3</th></tr> <tr><td>Cubes in bottom row</td><td>2</td><td>4</td><td>6</td></tr> <tr><td>Total cubes</td><td>2</td><td>6</td><td>12</td></tr> </table> a) Lucas chooses the staircase height. Which quantities in the table depend on his choice? b) When the height goes from $2$ to $3$, how much do the bottom row and the total change? c) Let $h$ be the height and $b$ the number of cubes in the bottom row. Write an equation relating $b$ and $h$. Use it to find the bottom row for a height of $4$.

Hints

- Which quantity does Lucas decide first? - Compare the two table columns for heights $2$ and $3$, one row at a time. - Check whether doubling the height would double the number in the bottom row.

Solution

1. The bottom-row count and the total cube count depend on the height Lucas chooses. 2. From height $2$ to height $3$, the bottom row changes from $4$ to $6$, an increase of $2$; the total changes from $6$ to $12$, an increase of $6$. 3. The bottom-row rule is $b=2h$. At height $4$, $b=2\times4=8$.

Answer

a) The number of cubes in the bottom row and the total number of cubes. b) Bottom row increases by $2$ cubes; total increases by $6$ cubes. c) $b=2h$; at height $4$, the bottom row has $8$ cubes.

Dependent and independent variables 8

A craft booth charges $\$2$ per bead strand. Let $s$ be the number of strands and $c$ be the total cost. Identify the variable roles and write the equation.

Hints

- Separate the chosen count from the amount that results. - Use the per-item amount to connect the two variables.

Solution

1. The number of strands $s$ is chosen, so it is independent. 2. The total cost $c$ depends on that choice, so it is dependent. 3. The equation is $c=2s$.

Answer

Independent: $s$ Dependent: $c$ Equation: $c=2s$

Dependent and independent variables 9

A reading plan starts with $12$ pages already completed and adds $8$ pages each day. Let $d$ be days and $p$ be total pages completed. Which variable depends on the other, and what is the equation?

Hints

- Identify the starting amount separately from the repeated change. - Write the total number of pages in terms of the elapsed days.

Solution

1. The number of days $d$ is the independent variable. 2. The total pages $p$ depend on the days. 3. Starting at $12$ and adding $8$ per day gives $p=12+8d$.

Answer

Independent: $d$ Dependent: $p$ Equation: $p=12+8d$

Dependent and independent variables 10

For the equation $y=5x$, suppose $x$ is the number of bags and $y$ is the number of apples. Explain which variable is independent and what the coefficient $5$ means.

Hints

- Use the context to decide which quantity is counted first. - Interpret the numerical factor as a change in the dependent quantity for one unit of the independent quantity.

Solution

1. The number of bags $x$ is selected, so it is independent. 2. The number of apples $y$ depends on the bag count. 3. The coefficient $5$ means there are $5$ apples per bag.

Answer

Independent variable: $x$ Dependent variable: $y$ The $5$ means $5$ apples per bag.

Dependent and independent variables 11

A candle is $18$ centimeters tall and burns $2$ centimeters each hour. Let $h$ be hours and $r$ be the remaining height. Identify the variables and write an equation.

Hints

- Determine which quantity progresses and which quantity responds. - Use subtraction because the output decreases from a starting amount.

Solution

1. Time $h$ is independent. 2. Remaining height $r$ depends on time. 3. The candle starts at $18$ and decreases by $2$ per hour, so $r=18-2h$.

Answer

Independent: $h$ Dependent: $r$ Equation: $r=18-2h$

Dependent and independent variables 12

The table records a relationship. <table> <thead> <tr> <th>Number of trays $t$</th> <th>Number of cups $c$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$6$</td> </tr> <tr> <td>$2$</td> <td>$12$</td> </tr> <tr> <td>$4$</td> <td>$24$</td> </tr> </tbody> </table> Which variable is independent, which is dependent, and what equation fits the table?

Hints

- Look at which column lists the chosen counts and which lists resulting totals. - Compare each output with its input to find the consistent relationship.

Solution

1. The tray count $t$ is the input and is independent. 2. The cup count $c$ changes with the tray count and is dependent. 3. Each tray corresponds to $6$ cups, so $c=6t$.

Answer

Independent: $t$ Dependent: $c$ Equation: $c=6t$

Dependent and independent variables 13

Thabo writes $d=4t$ for distance $d$ traveled in $t$ hours. Yuna says $t$ must be dependent because it is on the right side of the equation. Evaluate Yuna's claim.

Hints

- Use the causal or input-output relationship in the context. - Do not decide variable roles from left-side and right-side placement alone.

Solution

1. Variable roles come from the situation, not merely from equation position. 2. Elapsed time $t$ is treated as the input. 3. Distance $d$ depends on time, so $t$ is independent and $d$ is dependent.

Answer

Yuna's claim is incorrect. $t$ is independent and $d$ is dependent.

Dependent and independent variables 14

A recipe uses $\frac{3}{4}$ cup of oats per batch. Let $b$ be batches and $o$ be cups of oats. Write $o$ in terms of $b$ and identify both variable roles.

Hints

- Use the number of batches as the repeated-group count. - Multiply the amount per batch by the input count.

Solution

1. The batch count $b$ is independent. 2. The oat amount $o$ depends on the number of batches. 3. The equation is $o=\frac{3}{4}b$.

Answer

Independent: $b$ Dependent: $o$ Equation: $o=\frac{3}{4}b$

Dependent and independent variables 15

A parking meter starts with $30$ minutes and adds $15$ minutes for each quarter inserted. Let $q$ be quarters and $m$ be total minutes. Write the equation and identify the dependent variable.

Hints

- Separate the initial amount from the amount added repeatedly. - The output is the quantity that changes after the input count is chosen.

Solution

1. The quarter count $q$ is independent. 2. Total minutes $m$ depend on the number of quarters. 3. The equation is $m=30+15q$.

Answer

Equation: $m=30+15q$ Dependent variable: $m$

Dependent and independent variables 16

A square has side length $s$ and area $A$. State which variable is independent and write the equation relating them.

Hints

- The geometric measurement chosen first determines the resulting area. - Use the standard area relationship for a square.

Solution

1. The side length $s$ is selected, so it is independent. 2. The area $A$ depends on the side length. 3. The equation is $A=s^2$.

Answer

Independent: $s$ Dependent: $A$ Equation: $A=s^2$

Dependent and independent variables 17

A roll begins with $50$ tickets. After $s$ tickets are sold, $r$ tickets remain. Write an equation and identify the dependent variable.

Hints

- Start from the original total and represent what is removed. - The remaining amount changes in response to the sold amount.

Solution

1. The sold count $s$ is treated as independent. 2. The remaining count $r$ depends on how many are sold. 3. The equation is $r=50-s$.

Answer

Equation: $r=50-s$ Dependent variable: $r$

Dependent and independent variables 18

A science class cools a sample by $1.5$ degrees each minute from an initial $24$ degrees Celsius. Let $m$ be minutes and $T$ be temperature. Write the equation and identify the independent variable.

Hints

- Identify the initial measurement and the repeated change. - Elapsed time is the input that determines the later measurement.

Solution

1. Time $m$ is independent. 2. Temperature $T$ depends on elapsed time. 3. The sample starts at $24$ and decreases by $1.5$ per minute, so $T=24-1.5m$.

Answer

Equation: $T=24-1.5m$ Independent variable: $m$

Dependent and independent variables 19

A train moves at $45$ miles per hour. Choose variables, define them, and write an equation that gives distance in terms of time. Identify the variable roles.

Hints

- Choose variable names that match the two quantities. - Write the output quantity as the rate times the input quantity.

Solution

1. Let $t$ be time in hours and $d$ be distance in miles. 2. Time $t$ is independent, and distance $d$ is dependent. 3. At $45$ miles per hour, $d=45t$.

Answer

Let $t$ be time in hours and $d$ be distance in miles. Independent: $t$ Dependent: $d$ Equation: $d=45t$

Dependent and independent variables 20

A school store’s total price $P$ for $n$ folders is $P=1.25n$. What changes when $n$ increases by $1$, and which variable is responsible for that change?

Hints

- Compare the equation for two consecutive input values. - Connect the coefficient to the change in the output for one added item.

Solution

1. The input $n$ increases by $1$. 2. The equation adds another $1.25$ to $P$. 3. Thus the dependent variable $P$ increases by $\$1.25$, caused by the change in independent variable $n$.

Answer

When $n$ increases by $1$, $P$ increases by $\$1.25$. The independent variable $n$ drives the change.

Dependent and independent variables 21

A photo printer takes $4$ seconds per photo. Which equation correctly gives total time $t$ in terms of photo count $p$: $p=4t$, $t=4p$, or $t=p+4$? Identify the variable roles.

Hints

- Test each equation using one photo. - The dependent quantity should equal the per-item amount times the count.

Solution

1. The photo count $p$ is selected, so it is independent. 2. Total time $t$ depends on that count. 3. Four seconds for each photo gives $t=4p$.

Answer

Correct equation: $t=4p$ Independent: $p$ Dependent: $t$

Dependent and independent variables 22

Samira chooses how many hours to practice, and the table shows how many songs she learns after each amount of practice. <table> <thead> <tr> <th>Hours $h$</th> <th>Songs $s$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$2$</td> </tr> <tr> <td>$2$</td> <td>$4$</td> </tr> <tr> <td>$3$</td> <td>$6$</td> </tr> </tbody> </table> Samira claims songs should be independent because songs appear in the right column. Explain the error.

Hints

- Use the meaning of the quantities rather than their display order. - Ask which quantity is chosen first and which quantity changes in response.

Solution

1. Table position does not determine variable roles. 2. Samira chooses the practice time $h$, so it is the input. 3. Songs learned $s$ depend on the hours practiced. 4. Therefore, $h$ is independent and $s$ is dependent.

Answer

Samira's claim is incorrect. Hours $h$ are independent, and songs $s$ are dependent; column position alone does not decide the roles.

Dependent and independent variables 23

A delivery drone’s height $H$ after $t$ seconds is modeled by $H=2t+6$. Identify the starting height, the independent variable, and the dependent variable.

Hints

- Look at the term that remains when the input is zero. - Use the context to identify which measurement changes as time passes.

Solution

1. At $t=0$, $H=6$, so the starting height is $6$. 2. Time $t$ is independent. 3. Height $H$ depends on time and is dependent.

Answer

Starting height: $6$ Independent variable: $t$ Dependent variable: $H$

Dependent and independent variables 24

A rectangular garden has a fixed width of $4$ feet and a variable length $l$. Let $A$ be area. Write the relationship and identify the dependent quantity.

Hints

- Use the fixed dimension with the variable dimension in the area relationship. - The measurement calculated from the chosen length is the dependent quantity.

Solution

1. The length $l$ is the independent variable. 2. Area is width times length. 3. With width $4$, $A=4l$. 4. The area $A$ is dependent.

Answer

Equation: $A=4l$ Dependent variable: $A$

Dependent and independent variables 25

In each situation, identify the dependent variable. Use the situation to decide which quantity changes as a result of the other; do not use equation-side position as a rule. a) A store chooses the number of pencil packs $x$ to order, and the total number of pencils $y$ is described by $7x=y$. b) A game charges a fixed fee plus an amount for each round. The number of rounds $n$ is chosen first, and the total cost $C$ satisfies $C=10+3n$. c) Maya starts with $\$40$. The amount she spends $s$ is chosen, and the amount remaining $r$ satisfies $40-r=s$.

Hints

- For each situation, identify which quantity is chosen or controlled first. - The dependent variable is the quantity whose value follows from that choice. - Check the context rather than relying on which side of the equals sign contains a variable.

Solution

1. In a), the number of ordered packs $x$ determines the total number of pencils, so $y$ is dependent. 2. In b), the chosen number of rounds $n$ determines the total cost, so $C$ is dependent. 3. In c), the chosen amount spent $s$ determines the amount remaining, so $r$ is dependent. 4. The dependent variable appears on different sides or in different positions in the three equations, so equation position is not the deciding feature.

Answer

a) $y$ b) $C$ c) $r$

Dependent and independent variables 26

Matías can choose the number $k$ of laps to swim. Total distance $d$ is then calculated. Which variable belongs on the x-axis of a graph, and why?

Hints

- Identify which quantity is chosen and which quantity depends on it. - Recall how input and output variables are normally assigned to graph axes.

Solution

1. The lap count $k$ is chosen and is independent. 2. The independent variable is conventionally placed on the x-axis. 3. Therefore, $k$ belongs on the x-axis.

Answer

$k$ belongs on the x-axis because it is the independent variable.

Dependent and independent variables 27

A theater arranges $r$ rows of seats, with $24$ seats in each row. At least one row is used. Let $s$ be the total number of seats. Which variable is independent? What numbers of rows make sense? Write an equation for $s$.

Hints

- Does this seating plan allow part of a row? - Which amount determines the other? - How many seats are added for every new row?

Solution

1. The theater chooses the number of rows, so $r$ is independent. 2. The number of rows must be a positive whole number. 3. The number of seats depends on the rows: $s=24r$.

Answer

Independent variable: $r$ Possible row counts: positive whole numbers $1,2,3,\dots$ Equation: $s=24r$

Dependent and independent variables 28

A classroom has $5$ tables with $c$ chairs at each table, so total chairs are $T=5c$. Which quantity may vary in this equation, and which variable is dependent?

Hints

- Distinguish the fixed number from the quantities represented by variables. - The total changes when the per-table amount changes.

Solution

1. The number of tables is fixed at $5$. 2. Chairs per table $c$ may vary and is the independent variable. 3. Total chairs $T$ depend on $c$, so $T$ is dependent.

Answer

The varying independent quantity is $c$, chairs per table. The dependent variable is $T$, total chairs.

Dependent and independent variables 29

A graphing program asks for the independent variable first. For the relationship $M=7+0.5d$, where $d$ is days and $M$ is plant mass in grams, which variable should be entered first and what does $7$ represent?

Hints

- Identify the input from the context before using the equation. - Interpret the constant by considering a zero input.

Solution

1. Days $d$ are the independent variable, so $d$ should be entered first. 2. Mass $M$ is dependent. 3. At $d=0$, $M=7$, so $7$ is the initial mass in grams.

Answer

Enter $d$ first. The $7$ represents the initial mass of $7\,\text{g}$.

Dependent and independent variables 30

Anouk says an independent variable never changes. Correct the statement using the relationship $C=3n$, where $n$ is item count and $C$ is cost.

Hints

- Distinguish “independent” from “fixed.” - Use several possible item counts to see how the input can vary.

Solution

1. An independent variable may take different chosen values. 2. In $C=3n$, the item count $n$ can change from one case to another. 3. “Independent” means it is treated as the input, not that it is constant. 4. Cost $C$ depends on the chosen value of $n$.

Answer

Anouk's statement is false. An independent variable can change; it is the input whose chosen values determine the dependent variable.

Dependent and independent variables 31

A water tank is draining. In the table, $t$ is time in hours and $W$ is water left in gallons. <table><tr><th>$t$</th><th>$0$</th><th>$1$</th><th>$2$</th></tr><tr><th>$W$</th><td>$20$</td><td>$17$</td><td>$14$</td></tr></table> Which variable is independent? Which is dependent? What does a change of $1$ in the independent variable mean here?

Hints

- Which quantity can you choose first to predict the other one? - Compare the water amounts at $t=0$ and $t=2$. What changed as time passed?

Solution

1. Time $t$ is the input, so it is independent; the remaining water $W$ is determined by elapsed time, so it is dependent. 2. An increase of $1$ in $t$ means one more hour has passed, leaving $3$ fewer gallons.

Answer

Independent: $t$, time in hours. Dependent: $W$, remaining gallons. One more hour means $3$ fewer gallons.

Dependent and independent variables 32

A gardener uses the graph to predict a plant's growth. A friend says height is independent because it increases. Is that correct? Explain which value must be chosen first. How much taller is the plant predicted to be at week $3$ than at week $1$?

Hints

- Does a quantity become independent merely because its numbers increase? - Which number must the gardener pick before reading a prediction? - Read the two heights and use them to support your explanation.

Solution

1. The claim is incorrect: the gardener selects an age in weeks to predict a height. Age is the independent quantity; predicted height depends on age. 2. At week $1$, the predicted height is $6$ centimeters; at week $3$, it is $10$ centimeters. The predicted increase is $10-6=4$ centimeters.

Answer

No. Age in weeks is independent because the gardener chooses an age to make a prediction; height depends on age. The plant is predicted to be $4\,\text{cm}$ taller at week $3$ ($10\,\text{cm}$) than at week $1$ ($6\,\text{cm}$).

Dependent and independent variables 33

A snack mix contains $10$ scoops in all. Let $x$ be the number of fruit scoops and $y$ be the number of cereal scoops, so $x+y=10$. a) Give three ordered pairs of positive whole numbers $(x, y)$ that satisfy the equation. b) If $x$ is chosen first, explain why $y$ is the dependent variable.

Hints

- Use the fixed total to check each ordered pair. - Keep both coordinates positive whole numbers. - Ask which value is determined after the other is chosen.

Solution

1. Positive whole-number pairs whose coordinates add to $10$ include $(1, 9)$, $(4, 6)$, and $(7, 3)$. 2. Once $x$ is chosen, $y$ must equal $10-x$. Therefore, the value of $y$ depends on the chosen value of $x$.

Answer

a) Sample pairs: $(1, 9)$, $(4, 6)$, and $(7, 3)$ b) $y$ is dependent because $y=10-x$ once $x$ is chosen.

Dependent and independent variables 34

A teacher divides $24$ markers equally among $x$ groups. Let $T$ be the number of markers in each group, so $T=24\div x$. a) Find $T$ when $x=2$, $x=4$, and $x=8$. b) Identify the independent and dependent variables. c) Describe how $T$ changes for these values as $x$ increases.

Hints

- Substitute each number of groups into $T=24\div x$. - Decide which quantity is chosen and which results from that choice. - Compare the three outputs in order.

Solution

1. When $x=2$, $T=24\div2=12$. When $x=4$, $T=24\div4=6$. When $x=8$, $T=24\div8=3$. 2. The number of groups $x$ is independent because it is chosen first. The number in each group $T$ depends on $x$. 3. For the given values, increasing the number of groups decreases the number of markers in each group.

Answer

a) $T=12$, $6$, and $3$, respectively b) Independent: $x$; dependent: $T$ c) $T$ decreases as $x$ increases for the given values.

Dependent and independent variables 35

Maya makes a chain of triangles with craft sticks. The first triangle needs $3$ sticks. Each additional triangle needs only $2$ more sticks because neighboring triangles share one side. a) How many sticks are needed for a chain of $25$ triangles? b) Maya uses all $61$ sticks in a box. How many triangles are in the chain? c) Let $n$ be the number of triangles and $s$ be the number of sticks. Write an equation that relates $s$ to $n$.

Hints

- Compare what changes when one more triangle is added to the chain. - For the reverse question, undo the same relationship that connects the number of triangles to the number of sticks. - In the equation, make the number of sticks depend on the number of triangles.

Solution

a) The first triangle uses $3$ sticks and the other $24$ triangles add $2$ sticks each, so $3 + 24 \times 2 = 51$ sticks. b) The relationship can be reversed: $(61 - 1) \div 2 = 30$, so the chain has $30$ triangles. c) Each triangle contributes two sticks to the repeating pattern, with one additional stick at the end. The equation is $s = 2n + 1$.

Answer

a) $51$ sticks b) $30$ triangles c) $s = 2n + 1$

Dependent and independent variables 36

Consider the expression $12.5-x$. a) Evaluate the expression for $x=2$, $x=5$, and $x=10$. b) How does the value of the expression change as $x$ increases through these values? Explain.

Hints

- Substitute one value of $x$ at a time. - Compare the three results in the same order as the inputs. - Think about subtracting increasingly large amounts from a fixed number.

Solution

1. For $x=2$, $12.5-2=10.5$. 2. For $x=5$, $12.5-5=7.5$. 3. For $x=10$, $12.5-10=2.5$. 4. As $x$ increases from $2$ to $5$ to $10$, the value decreases from $10.5$ to $7.5$ to $2.5$. 5. A greater amount is being subtracted from the same starting value, so the result becomes smaller.

Answer

a) For $x=2$, the value is $10.5$; for $x=5$, it is $7.5$; for $x=10$, it is $2.5$. b) The value decreases as $x$ increases.

Dependent and independent variables 37

A bicycle rack holds $2$ wheels for each bicycle. Let $b$ be bicycles and $w$ be wheels. Marek writes $b=2w$. Correct the equation and identify the variable roles.

Hints

- Check the equation with one bicycle as a simple test case. - Place the resulting total in terms of the chosen count.

Solution

1. The bicycle count $b$ is independent. 2. The wheel count $w$ is twice the bicycle count. 3. The correct equation is $w=2b$. 4. Marek reversed which quantity is multiplied by $2$.

Answer

Independent: $b$ Dependent: $w$ Correct equation: $w=2b$

Dependent and independent variables 38

The variable-role headings in the table are incorrect. <table> <thead> <tr> <th>Independent: total cost $c$</th> <th>Dependent: number of markers $n$</th> </tr> </thead> <tbody> <tr> <td>$3$</td> <td>$1$</td> </tr> <tr> <td>$6$</td> <td>$2$</td> </tr> <tr> <td>$9$</td> <td>$3$</td> </tr> </tbody> </table> Markers cost $\$3$ each. Correct the variable-role headings.

Hints

- Use the pricing situation rather than trusting the printed headings. - Ask which quantity can be selected before calculating the other.

Solution

1. The number of markers is chosen first. 2. The total cost changes according to that count. 3. Therefore, $n$ is independent and $c$ is dependent.

Answer

Independent variable: number of markers $n$ Dependent variable: total cost $c$

Dependent and independent variables 39

A tank contains $80$ liters and drains $5$ liters each minute. Complete the statement: “The amount remaining depends on ______ because ______.” Then write the equation using $m$ for minutes and $L$ for liters remaining.

Hints

- Explain the direction of dependence in words before writing symbols. - Represent the fixed start and the repeated loss separately.

Solution

1. The amount remaining changes according to elapsed time. 2. Therefore, liters remaining depend on minutes because time determines how much has drained. 3. The equation is $L=80-5m$.

Answer

The amount remaining depends on minutes because elapsed time determines how much has drained. $L=80-5m$

Dependent and independent variables 40

For $P=4s$, where $s$ is the side length of a square and $P$ is its perimeter, compare what happens to $P$ when $s$ changes from $3$ to $5$. Which variable was changed directly?

Hints

- Evaluate the relationship at both input values. - Distinguish the value selected directly from the value recalculated afterward.

Solution

1. At $s=3$, $P=4\times3=12$. 2. At $s=5$, $P=4\times5=20$. 3. The perimeter increases by $8$. 4. The side length $s$, the independent variable, was changed directly.

Answer

$P$ changes from $12$ to $20$, an increase of $8$. The directly changed variable is $s$.

Dependent and independent variables 41

A weather station records time $t$ and air temperature $T$. Can the station choose the temperature first and make time depend on it in this data collection? Explain the intended variable roles.

Hints

- Use the design of the data collection to identify the input. - Variable roles can depend on how a relationship is being studied.

Solution

1. In this observation plan, time is the scheduled input for each measurement. 2. The recorded temperature is observed at each time. 3. Therefore, $t$ is independent and $T$ is dependent in the data relationship.

Answer

In this data collection, $t$ is independent and $T$ is dependent because temperature is recorded at selected times.

Dependent and independent variables 42

A ferry charges $\$6$ to board plus $\$2$ per bicycle. Let $b$ be bicycles and $C$ be total cost. Write the equation, identify the variables, and state the meaning of $C$ when $b=0$.

Hints

- Separate the fixed amount from the amount tied to the input count. - Interpret the output when the independent variable is zero.

Solution

1. Bicycle count $b$ is independent, and cost $C$ is dependent. 2. The equation is $C=6+2b$. 3. At $b=0$, $C=\$6$, representing the boarding charge with no bicycles.

Answer

Independent: $b$ Dependent: $C$ Equation: $C=6+2b$ $C=\$6$ when $b=0$, the boarding charge alone.

Dependent and independent variables 43

A craft-stick display uses $6$ sticks for the first section and $4$ additional sticks for each later section. Let $n$ be the number of sections and $s$ be the number of sticks. Which variable is independent, and which equation fits: $s=4n+2$ or $s=6n+4$? Justify the equation from the pattern description.

Hints

- Separate the sticks for the first section from the sticks added by the remaining sections. - The number of later sections is one less than the total number of sections.

Solution

1. The number of sections $n$ is independent, and the number of sticks $s$ is dependent. 2. The first section uses $6$ sticks, and the remaining $n-1$ sections add $4$ sticks each. 3. Thus $s=6+4(n-1)=6+4n-4=4n+2$. 4. Therefore, the matching equation is $s=4n+2$.

Answer

Independent: $n$ Dependent: $s$ Equation: $s=4n+2$, because $6+4(n-1)=4n+2$.

Dependent and independent variables 44

The equation $a=12-p$ relates available seats $a$ to occupied seats $p$ in a $12$-seat van. If $p$ is chosen as the independent variable, explain how $a$ changes when $p$ increases by $1$.

Hints

- Compare the output before and after increasing the input by one. - The subtraction sign indicates the two quantities change in opposite directions.

Solution

1. The variable $a$ is dependent on $p$. 2. Replacing $p$ by $p+1$ subtracts one additional seat from $12$. 3. Therefore, $a$ decreases by $1$ when $p$ increases by $1$.

Answer

The dependent variable $a$ decreases by $1$ for each increase of $1$ in $p$.

Dependent and independent variables 45

A baker records oven time $t$ and number of loaves $L$ baked. The equation $L=3t$ is proposed, with $t$ in hours. What assumption about the variable relationship does this equation make? Identify the variable roles.

Hints

- Interpret what one unit of the input does to the output. - Notice what the equation predicts when the input is zero.

Solution

1. Time $t$ is independent, and loaves $L$ are dependent. 2. The coefficient $3$ means the model assumes $3$ loaves are baked per hour. 3. The equation also assumes the relationship continues at that constant rate from $0$.

Answer

Independent: $t$ Dependent: $L$ The equation assumes a constant rate of $3$ loaves per hour starting from $0$.

Dependent and independent variables 46

A table for $y=2x+1$ lists the pair $(4, 9)$. If $x$ is independent, explain why writing the pair as $(9, 4)$ changes its meaning.

Hints

- Use the input-output order of coordinates. - Check the reversed pair in the equation to see whether it still fits.

Solution

1. Ordered pairs list the independent value first and dependent value second. 2. The pair $(4, 9)$ means $x=4$ gives $y=9$. 3. The reversed pair $(9, 4)$ would claim $x=9$ gives $y=4$, which does not satisfy the equation.

Answer

$(4, 9)$ means input $4$ gives output $9$. Reversing it swaps the variable roles and produces a point that does not satisfy $y=2x+1$.

Dependent and independent variables 47

A runner moves at a constant $6$ miles per hour. Elapsed time $t$, in hours, is the input, and distance $d$, in miles, is calculated by $d=6t$. Luka says doubling $d$ will always cause $t$ to double, so $d$ should be independent. Explain why that conclusion about variable roles does not follow.

Hints

- Separate a numerical pattern from the definition of input and output in the model. - Use the stated way the relationship is generated to assign roles.

Solution

1. The equation does imply corresponding doubled values in this relationship. 2. However, variable roles are set by the model: elapsed time $t$ is the input. 3. Distance $d$ is calculated from time and remains the dependent variable.

Answer

The doubling relationship does not determine the roles. In this model, $t$ is the independent input and $d$ is the dependent output.

Dependent and independent variables 48

Jordan wrote $P=4g$ for the arcade relationship. Read the graph at $g=0$, then use the marked points at $g=2$ and $g=4$ to find the change in points per game. What did Jordan leave out? Write the correct equation.

Hints

- Look for the marked point where the number of games is zero. - Compare the vertical and horizontal changes between the other two marked points. - The equation must reflect both the starting amount and the change per game.

Solution

1. At $g=0$, the graph shows $P=6$. 2. From $(2,14)$ to $(4,22)$, points increase by $8$ over $2$ games, or $4$ points per game. 3. Jordan left out the starting $6$ points. The equation is $P=4g+6$.

Answer

Starting points: $6$; increase per game: $4$. Jordan omitted the starting amount. Correct equation: $P=4g+6$.

Dependent and independent variables 49

A square flower bed is surrounded by one row of square paving stones. The side length of the bed is measured in stone lengths. <table> <tr><td>Side length of flower bed</td><td>$1$</td><td>$2$</td><td>$3$</td></tr> <tr><td>Number of paving stones</td><td>$8$</td><td>$12$</td><td>$16$</td></tr> </table> a) How many paving stones are needed when the side length is $12$? b) Let $n$ be the side length and $p$ be the number of paving stones. Write an equation that relates $p$ to $n$.

Hints

- Compare consecutive rows of the table to identify how the paving-stone count changes. - Relate the changing paving-stone count to the changing side length. - In part b), write the paving-stone count as the dependent quantity.

Solution

a) The number of paving stones increases by $4$ whenever the side length increases by $1$. For side length $12$, $4 \times 12 + 4 = 52$, so $52$ paving stones are needed. b) Four side lengths contribute $4n$, with $4$ additional corner stones. The equation is $p = 4n + 4$.

Answer

a) $52$ paving stones b) $p = 4n + 4$

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Grade 6 math worksheets

Dependent and independent variables

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