Dependent and Independent Variables: practice solutions, Grade 6 – download the PDF

1 Name the variables ★★★

1. The calories burned depend on the time, so the independent variable is the minutes and the dependent variable is the calories.
2. The price depends on how many pizzas you order: independent = number of pizzas, dependent = total price.
3. The level in the barrel depends on the rain: independent = amount of rain, dependent = water level.

2 Complete the table ★★★

Multiply each \( x \) by 4: \( 4 \times 1 = 4 \), \( 4 \times 2 = 8 \), \( 4 \times 3 = 12 \), \( 4 \times 4 = 16 \), \( 4 \times 5 = 20 \).

3 Movie tickets ★★★

1. Each ticket costs 9 dollars, so \( c = 9t \).
2. \( c = 9 \times 6 = 54 \). Six tickets cost 54 dollars.
3. \( 9t = 117 \), so \( t = 117 \div 9 = 13 \). The group bought 13 tickets.

4 Use the rule ★★★

Add 7 to each input: \( 3 + 7 = 10 \), \( 8 + 7 = 15 \), \( 15 + 7 = 22 \). So \( y = 10 \), \( 15 \) and \( 22 \).

5 From a table to ordered pairs ★★★

The pairs are \( (1, 2) \), \( (2, 4) \) and \( (3, 6) \). The independent variable (hours) comes first and goes on the horizontal axis. The dependent variable (pages) comes second and goes on the vertical axis.

6 True or false? ★★★

1. False. \( d \) is found by multiplying \( t \) by 5, so \( d \) depends on \( t \): it is the dependent variable.
2. True. In \( (4, 20) \), \( x = 4 \) is the input and \( y = 20 = 5 \times 4 \) is the output.
3. True. This is the convention: the independent variable on the horizontal axis.
4. False. \( y = 18 + 3 = 21 \). The value \( y = 6 \) goes with \( x = 3 \).

7 Growing bamboo ★★★

1. \( d \)	1	2	3	4	5
   \( h \)	2	4	6	8	10

2. \( h = 2d \).
3. \( h = 2 \times 7 = 14 \) inches. In centimeters, \( 14 \times 2.54 = 35.56 \), about 35.6 cm.

8 Find the rule: adding ★★★

The outputs go up by 1 as the inputs go up by 1. Since \( 9 = 2 + 7 \), \( 10 = 3 + 7 \), \( 11 = 4 + 7 \) and \( 12 = 5 + 7 \), the rule is \( y = x + 7 \). For \( x = 40 \), \( y = 47 \).

9 Find the rule: multiplying ★★★

The outputs go up by 6 each time and \( y \) is 6 times \( x \) in every column, so \( y = 6x \). For \( x = 10 \), \( y = 60 \). For \( y = 96 \), \( 6x = 96 \), so \( x = 16 \).

10 Reading a graph ★★★

1. The savings depend on the number of weeks. Independent: \( w \). Dependent: \( s \).
2. The points go up by 4 for each week: 4 dollars per week.
3. \( s = 4w \).
4. \( s = 4 \times 8 = 32 \). After 8 weeks she will have saved 32 dollars.

11 Cousins ★★★

1. \( L = a + 4 \).

2. \( a \)	6	9	12
   \( L \)	10	13	16

3. \( a + 4 = 20 \), so \( a = 16 \). The cousin is 16.
4. No. \( 10 : 6 \) and \( 13 : 9 \) are different ratios (\( \dfrac{10}{6} \approx 1.67 \) and \( \dfrac{13}{9} \approx 1.44 \)), so the relationship is not a constant ratio.

12 The jogger ★★★

1. Each 5 minutes adds 0.5 mile, so the rate is \( \dfrac{0.5}{5} = 0.1 \) mile per minute (that is 6 miles per hour).
2. \( d = 0.1m \).
3. \( d = 0.1 \times 45 = 4.5 \). She runs 4.5 miles, about 7.2 kilometers.

13 Is the rate constant? ★★★

The cost goes up by \( 6 - 3 = 3 \), then \( 10 - 6 = 4 \), then \( 15 - 10 = 5 \). The change is not always the same, so the rate is not constant. If \( c = kn \), then \( k = 3 \) from the first pair, but \( 3 \times 3 = 9 \ne 10 \). So the relationship cannot be written as \( c = kn \).

14 Trail mix ratio ★★★

1. \( n \)	2	4	6	8
   \( r \)	3	6	9	12

2. \( (2, 3) \), \( (4, 6) \), \( (6, 9) \), \( (8, 12) \).
3. Yes. The ratio \( 3 : 2 \) is the same for every pair and \( (0, 0) \) fits, so the points are on a line through the origin.
4. The rule is \( r = \dfrac{3}{2}n \). For \( n = 10 \), \( r = \dfrac{3}{2} \times 10 = 15 \) cups.

15 Taxi fare ★★★

1. \( c = 2m + 3 \).

2. \( m \)	0	1	2	3	4	5
   \( c \)	3	5	7	9	11	13

3. The cost goes up by 2 dollars for each extra mile, so the constant rate is 2 dollars per mile. The 3 dollars is the starting fee, not part of the rate.
4. \( 2m + 3 = 25 \), so \( 2m = 22 \) and \( m = 11 \). The ride was 11 miles (about 17.7 km).

16 Two cyclists ★★★

1. Ana. Her line is steeper: she reaches 60 miles at \( t = 5 \) while Ben reaches 45 miles.
2. Ana: \( d = 12t \). Ben: \( d = 9t \).
3. Ana: \( 12 \times 3 = 36 \) miles. Ben: \( 9 \times 3 = 27 \) miles. The difference is \( 36 - 27 = 9 \) miles.
4. The gap is \( 12t - 9t = 3t \). We need \( 3t = 12 \), so \( t = 4 \) hours.

17 Find the mistake ★★★

The rule works for \( x = 1 \) because \( 1 + 3 = 4 \). But for \( x = 2 \), \( 2 + 3 = 5 \ne 8 \). A rule must work for every pair. Here \( y \) is 4 times \( x \) each time, so the correct rule is \( y = 4x \). For \( x = 25 \), \( y = 100 \).

18 The printer ★★★

1. \( p = 18m \).
2. \( p = 18 \times 10 = 180 \) pages.
3. \( 18m = 135 \), so \( m = 135 \div 18 = 7.5 \). It takes 7.5 minutes.
4. The number of pages printed depends on the time, so \( m \) is independent and \( p \) is dependent.

19 A line that misses the origin ★★★

1. The pair is \( (0, 4) \). The line starts at height 4 on the vertical axis, not at the origin.
2. \( y \) goes up by 1 for each step of 1 in \( x \), so the rate is 1.
3. At \( x = 1 \), \( \dfrac{y}{x} = \dfrac{5}{1} = 5 \). At \( x = 4 \), \( \dfrac{y}{x} = \dfrac{8}{4} = 2 \). The ratios differ, so \( y : x \) is not constant.
4. A constant ratio would give \( y = kx \), which always passes through \( (0, 0) \). This line passes through \( (0, 4) \) instead.

20 Your own situation ★★★

Sample answer. A baker packs 6 muffins in each box. Let \( b \) be the number of boxes (independent) and \( m \) the number of muffins (dependent). The equation is \( m = 6b \).

Any situation with 6 items per group works, as long as the table follows the same rule.

21 A road trip ★★★

1. \( d = 55t \).
2. \( d = 55 \times 3.5 = 192.5 \) miles.
3. \( 55t = 330 \), so \( t = 330 \div 55 = 6 \) hours.
4. \( 55 \times 1.609 = 88.495 \), about 88.5 kilometers per hour.

22 Missing values ★★★

For \( y = 35 \): \( 7x = 35 \), so \( x = 5 \). For \( x = 10 \): \( y = 70 \). For \( y = 63 \): \( 7x = 63 \), so \( x = 9 \). Check the first column: \( 7 \times 2 = 14 \). The missing values are 5, 70 and 9.