# Everyday Math Grade 6 Answers Unit 7 Variables and Algebraic Relationships

## Everyday Mathematics 6th Grade Answer Key Unit 7 Variables and Algebraic Relationships

Mystery Numbers
Question 1.
Gabe and Aurelia play Number Squeeze.
Gabe represents his mystery number with the variable f.
a. Represent each of the two Number Squeeze clues with an inequality. Describe the solution sets to the inequalities.

The given clues are:
a) Subtract 5 from f and the answer is greater than 7
So,
The representation of inequality and the solution set for the given clue is:
Inequality:
f – 5 > 7
Solution set:
f – 5 + 5 > 7 + 5
f > 12
So,
The solution set is: {All the numbers greater than 12}
b) The number f is less than 13
So,
The representation of inequality and the solution set for the given clue is:
Inequality:
f < 13
Solution set:
The solution set is: {All the numbers less than 13}
Hence, from the above,
The completed table is:

b. Graph the solution set that makes both inequalities true.

From part (a),
The inequalities are:
f > 12 and f < 13
Hence,
The graph of the  solution set that makes both inequalities true is:

c. List three numbers that could be the mystery number.
Check that they are in the solution sets for both inequalities.
Possible numbers f could be: ____________
From part (b),
We know that,
When we observe the graph of both the inequalities, the mystery numbers for both lie between 12 and 13 only
Hence, from the above,
We can conclude that
The possible numbers of f could be 12.1, 12.2,……………… 12.9

Question 2.
a. Write two inequalities that could be clues for the following graph:

Inequality A: ____________
Inequality B: ____________
The given graph is:

From the given graph,
We can observe that the numbers are between 3 and 7 with both  and 7 excluded
So,
The inequalities must be:
x > 3 and x < 7
Hence, from the above,
We can conclude that
Inequality A: x > 3
Inequality B: x < 7

b. Write a different set of inequalities that could also represent the graph in Problem 2a.
Inequality C: ____________
Inequality D: ____________
From part (a),
Inequality A: x > 3
Inequality B: x < 7
So,
The representation of inequality A and B in a different set of inequalities are:
Inequality C: Different set inequality of Inequality A
Inequality D: Different set Inequality of Inequality B
So,
Inequality C: x + 5 > 8
Inequality D: x – 4 < 3

Practice Evaluate.
Question 3.
|-4| = ________
We know that,
| x| = x for x > 0
| x| = -x for x < 0
Hence, from the above,
We can conclude that
| -4| = 4 or -4

Question 4.
|-0.5| = _____
We know that,
| x| = x for x > 0
| x| = -x for x < 0
Hence, from the above,
We can conclude that
| -0.5| = 0.5 or -0.5

Question 5.
|z| = 6; z = ______
The given expression is:
| z | = 6
We know that,
| x| = x for x > 0
| x| = -x for x < 0
Hence, from the above,
We can conclude that
z = 6 or -6

Solving Problems with Inequalities
Fast and Healthy sells bags of trail mix. Customers choose the ingredients to put in their trail mix. The bag is weighed at the checkout counter to determine the cost.
Fast and Healthy charges $5 per pound. They also sell granola bars for$1.50 each.
Question 1.
Li has $9 to spend on trail mix. How many pounds of trail mix can she buy? Let y be the number of pounds of trail mix. a. Inequality for the situation: ___________ Answer: It is given that Fast and Healthy charges$5 per pound and they also sell granola bars for $1.50 each It is also given that Li has$9 to spend on trail mix
Now,
Let y be the number of pounds of trail mix
So,
The total money charged by Fast and Healthy is: $5y pounds But, Li can only buy the trail mix that is less than or equal to$9
Hence, from the above,
We can conclude that the inequality for the above situation is:
5y ≤ 9

b. Solution set for y using set notation: ____________
From part (a),
The inequality for the given situation is:
5y ≤ 9
y ≤ $$\frac{9}{5}$$
y ≤ 1.8
We know that,
The money will not be negative
Hence, from the above,
We can conclude that the solution set for the given inequality is:
{All numbers should be greater than 0 and less than or equal to 1.8}

c. Inequalities for the values of y: _____________
From part (b),
The inequality for the given situation is:
y ≤ 1.8
We know that,
The money will not be negative
Hence, from the above,
We can conclude that the inequalities for the value of y are:
y > 0 and y ≤ 1.8

d. Graph the solution set for y that makes both inequalities true.

From part (c),
The set of inequalities for the values of y are:
y > 0 and y ≤ 1.8
Hence, from the above,
We can conclude that the graph for the solution set for the given values of y is:

Question 2.
The price for a plain smoothie is $2.00. Each additional ingredient costs$0.75. Li has $5. Let m be the number of ingredients. How many ingredients can Li add to a plain smoothie? a. Inequality for the situation: __________ Answer: It is given that the price for a plain smoothie is$2 and each additional ingredient costs $0.75 It is also given that Li has only$5
Now,
Let m be the number of additional ingredients
So,
The total cost of additional ingredients is: $0.75m So, The number of ingredients that Li can add to a plain smoothie is: 2 + 0.75m But, We know that Li can only spend$5
So,
2 + 0.75m ≤ 5
Subtract with 2 on both sides
2 + 0.75m – 2 ≤ 5 – 2
0.75m ≤ 3
Hence, from the above,
We can conclude that
The inequality for the given situation is:
0.75m ≤ 3

b. Solution set for ‘m’ using set notation: __________
From part (a),
The inequality for the given situation is:
0.75m ≤ 3
m ≤ $$\frac{3}{0.75}$$
m ≤ 4
We know that,
The number of ingredients will not be negative
Hence, from the above,
We can conclude that
The solution set for the given inequality is:
{All numbers greater than or equal to 0 and less than or equal to 4}

c. Inequalities for the whole number values of m: __________
From part (b),
The inequality for the given situation is:
m ≤ 4
We know that,
The “Whole numbers” are the numbers that start from 0
Hence, from the above,
We can conclude that
The whole number set for the inequality in the given situation is:
{0, 1, 2, 3, 4}

d. Graph the solution set for ‘m’ that makes both inequalities true.

From part (c),
The inequality for the given situation is:
m ≤ 4
Hence, from the above,
We can conclude that the graph for the solution set for ‘m’ that makes both inequalities true is:

Question 3.
Describe how the graph in Problem 2d represents the solution to the problem.
We know that,
The ingredients will not be the decimal numbers like 1.5, 2.5, etc
So,
Only the whole numbers will be the solution set for the problem 2d
Hence, from the graph present in problem 2d,
The solution set of the problem 2d is: {0, 1, 2, 3, 4}

Practice
Solve.
Question 4.
$$\frac{2}{3}$$ × _____ = 1
Let the missing number be x
So,
$$\frac{2}{3}$$ × x = 1
Multiply by $$\frac{3}{2}$$ on both sides
So,
$$\frac{2}{3}$$ × $$\frac{3}{2}$$ × x = 1 × $$\frac{3}{2}$$
x = $$\frac{3}{2}$$
Hence, from the above,
We can conclude that the value of x from the given expression is: $$\frac{3}{2}$$

Question 5.
______ × 5 = 1
Let the missing number be x
So,
x × 5 = 1
Divide by 5 on both sides
So,
x × $$\frac{5}{5}$$ = $$\frac{1}{5}$$
x = $$\frac{1}{5}$$
Hence, from the above,
We can conclude that the value of x from the given expression is: $$\frac{1}{5}$$

Question 6.
3 $$\frac{3}{4}$$ × ______ = 1
Let the missing number be x
So,
3$$\frac{3}{4}$$ × x = 1
We know that,
3$$\frac{3}{4}$$ = $$\frac{15}{4}$$
So,
$$\frac{15}{4}$$ × x = 1
Multiply by $$\frac{4}{15}$$ on both sides
So,
$$\frac{15}{4}$$ × $$\frac{4}{15}$$ × x = 1 × $$\frac{4}{15}$$
x = $$\frac{4}{15}$$
Hence, from the above,
We can conclude that the value of x from the given expression is: $$\frac{4}{15}$$

Question 1.
Complete the spreadsheet on the right. If you have a spreadsheet program at home, write formulas and use the “fill down” feature to do the calculations. If not, do the calculations yourself with a calculator.

Question 2.
Use the data in the spreadsheet to graph the number pairs for x and 2x on the first grid. Then graph the number pairs for x and 24 ⁄ x on the second grid. Connect the plotted points.

From part (a),

So,
For the 1st graph,
The ordered pairs (x, 2x) are:
(1, 2), (2, 4), (3, 6), (4, 8), (6, 12), (8, 16), (12, 24), (1.2, 2.4), (3.2, 6.4), (5, 10)
For the 2nd graph,
The ordered pairs (x, $$\frac{24}{x}$$) are:
(1, 12), (2, 6), (3,4), (4, 3), (6, 2), (8, 1.5), (12, 1), (1.2, 10), (3.2, 3.75), (5, 2.4)
Hence from the above,
We can conclude that the graphs are:

Question 4.
Describe two differences between the two graphs in Problem 2.
In Problem 2,
From the 1st graph,
We can observe that the graph is a straight line that increases as x increases
From the 2nd graph,
We can observe that the graph is a curve that is decreasing as x decreases

Practice
Find the GCF.
Question 4.
GCF (34, 42) = _______
We know that,
“GCF” is defined as the greatest common factor of the 2 numbers
So,
Factors of 42 and 34 are:
42 = 7 × 2 × 3
34 = 2 × 17
Hence, from the above,
We can conclude that
GCF (34, 42) = 2

Question 5.
GCF (49, 560) = ________
We know that,
“GCF” is defined as the greatest common factor of the 2 numbers
So,
Factors of 49 and 560 are:
49 = 7 × 7
560 = 7 × 5 × 4 × 4
Hence, from the above,
We can conclude that
GCF (49, 560) = 7

Question 6.
GCF (30, 75) = _______
We know that,
“GCF” is defined as the greatest common factor of the 2 numbers
So,
Factors of 30 and 75 are:
30 = 15 × 2
75 = 15 × 5
Hence, from the above,
We can conclude that
GCF (30, 75) = 15

Question 1.
Jenna has a large jar full of pennies, nickels, and dimes. She has 100 coins. She has 20 more nickels than pennies and half as many dimes as nickels. Enter formulas in the spreadsheet to calculate the number of coins and their value.

a. Coins: _________
b. Value of coins: _________
a)
It is given that Jenna has a large jar full of pennies, nickels, and dimes.
It is given that
The total number of coins that Jenna possessed = 100
So,
pennies + nickels + dimes = 100
It is also given that
She has 20 more nickels than pennies
So,
nickels – pennies = 20
It is also given that
Half as many dimes as nickels
So,
dimes = $$\frac{nickels}{2}$$
Now,
Let the number of dimes be: x
So,
2 dimes + dimes + 2 dimes – 20 = 100
5 dimes – 20 = 100
5 dimes = 100 + 20
5 dimes = 120
1 dime = $$\frac{120}{5}$$
1 dime = 24
We know that,
nickels = 2 (dimes)
= 2 (24)
= 48
pennies = nickels – 20
= 48 – 20
= 28
Hence, from the above,
We can conclude that
The number of coins are:
Nickels: 48
Dimes: 24
Pennies: 28
b)
We know that,
1 nickel = $0.05 1 dime =$0.10
1 penny = $0.01 So, The total value of the coins = 48 (0.05) + 24 (0.10) + 28 (0.01) =$5.08
Hence, from the above,
We can conclude that the total value of the coins is: $5.08 Question 2. To solve Problem 1a, why might you start with an even number of pennies? Answer: To solve problem 1a, We might have to start with an even number of pennies because you will divide by 2 for the number of dimes, which only come in whole quantities Question 3. What formula would you use to find the total value of the coins? Answer: The formula used to find the total number of coins is: The total number of coins =$0.05 (Number of nickels) + $0.10 (Number of dimes) +$0.01 (Number of pennies)

Question 4.
Use formulas to find the greatest four consecutive numbers whose sum is less than 1,000.

Practice
Divide.
Question 5.
$$\frac{7}{8}$$ ÷ $$\frac{1}{8}$$ = ______
The given expression is:
$$\frac{7}{8}$$ ÷ $$\frac{1}{8}$$
So,
$$\frac{7}{8}$$ ÷ $$\frac{1}{8}$$
= $$\frac{7}{8}$$ × $$\frac{8}{1}$$
= $$\frac{7 × 8}{8 × 1}$$
= $$\frac{7}{1}$$
= 7
Hence, from the above,
We can conclude that the value of the given expression is: 7

Question 6.
2 $$\frac{1}{4}$$ ÷ $$\frac{7}{8}$$ = ______
The given expression is:
2 $$\frac{1}{4}$$ ÷ $$\frac{7}{8}$$
We know that,
2$$\frac{1}{4}$$ = $$\frac{9}{4}$$
So,
$$\frac{9}{4}$$ ÷ $$\frac{7}{8}$$
=  $$\frac{9}{4}$$ × $$\frac{8}{7}$$
= $$\frac{9 × 8}{4 × 7}$$
= $$\frac{18}{7}$$
Hence, from the above,
We can conclude that the value of the given expression is: $$\frac{18}{7}$$

Question 7.
_____ = 1 $$\frac{2}{3}$$ ÷ 3
The given expression is:
1 $$\frac{2}{3}$$ ÷ 3
We know that,
1$$\frac{2}{3}$$ = $$\frac{5}{3}$$
So,
$$\frac{5}{3}$$ ÷ 3
=  $$\frac{5}{3}$$ × $$\frac{1}{3}$$
= $$\frac{5 × 1}{3 × 3}$$
= $$\frac{5}{9}$$
Hence, from the above,
We can conclude that the value of the given expression is: $$\frac{5}{9}$$

Which Activity Burns the Most Calories?
Question 1.
The amount of energy food will produce when it is digested by the body is measured in a unit called the calorie.
The table shows the number of calories used per minute and per hour by the average sixth-grader in Oakwood Junior High for various everyday activities. Complete the table. Use the information for Problems 2–3.

We know that,
1 hour = 60 minutes
So,
Calories/ Hour = (Calories / Minute) × 60
Calories / Minute = $$\frac{Calories / Hour}{60}$$
Hence,
The completed table for calorie usage by the sixth-grader is:

Question 2.
Kori spent 2 hours and 25 minutes doing one of the listed activities. He burned 145 calories. Which activity was he doing?

It is given that Kori spent 2 hours and 25 minutes doing one of the listed activities from the above table
So,
Now,
2 hours 25 minutes = 2 hours + 25 minutes
= 2 (60 minutes) + 25 minutes
= 120 minutes + 25 minutes
= 145 minutes
Now,
Divide 145 minutes by using the Calories / Minute values in the above table
The value for which we will get the whole number when we divide 145 by Calories / Minute, will be the activity
Hence, from the above,
We can conclude that the activity doing by Kori is: Watching TV

Question 3.
Kori sleeps about 8 $$\frac{1}{2}$$ hours per night and spends about 7 hours each school day eating, talking, and sitting. Does he burn more calories sleeping or at school? Explain.

It is given that,
Kori sleeps about 8$$\frac{1}{2}$$ hours per night and spends about 7 hours each school day eating
Now,
We know that,
8$$\frac{1}{2}$$ = $$\frac{17}{2}$$ hours
Now,
Calories / Hour obtained for sleeping for $$\frac{17}{2}$$ hours = $$\frac{17}{2}$$ × 42
= 357 Calories / Hour
Calories / Hour obtained for eating in school each day for 7 hours = 7 × 72
= 504 Calories / Hour
Hence, from the above,
We can conclude that Kori spends more calories at school

Question 4.
On Monday Edgar ran for 29 minutes and burned 270 calories. On Wednesday he biked for 25 minutes and burned 207 calories. On Friday he played soccer for 13 minutes and burned 124 calories. Which activity burns the most calories per minute? Explain how you know.
It is given that
On Monday Edgar ran for 29 minutes and burned 270 calories. On Wednesday he biked for 25 minutes and burned 207 calories. On Friday he played soccer for 13 minutes and burned 124 calories.
So,
The Calories / Minute burn by Edgar on Monday = $$\frac{Number of calories}{Number of Minutes}$$
= $$\frac{270}{29}$$
= 9.31 Calories / Minute
The Calories / Minute burn by Edgar on Wednesday = $$\frac{Number of calories}{Number of Minutes}$$
= $$\frac{207}{25}$$
= 8.28 Calories / Minute
The Calories / Minute burn by Edgar on Friday = $$\frac{Number of calories}{Number of Minutes}$$
= $$\frac{124}{13}$$
= 9.53 Calories / Minute
Hence, from the above,
We can conclude that soccer activity burns the most Calorie / Minute

Practice
Find the LCM.
Question 5.
LCM (12, 48) = _______
We know that,
“LCM” is the lowest common divisor of the 2 numbers that are evenly divisible by  both the numbers
Now,
The multiples of 12 are:
12 = 12, 24, 36, 48, 60
The multiples of 48 are:
48 = 48, 128, 144, 192, 240
Hence, from the above,
We can conclude that
LCM (12, 48) = 48

Question 6.
LCM (14, 21) = _________
We know that,
“LCM” is the lowest common divisor of the 2 numbers that are evenly divisible by  both the numbers
Now,
The multiples of 14 are:
14 = 14, 28, 42, 56, 70
The multiples of 21 are:
21 = 21, 42, 63, 84, 105
Hence, from the above,
We can conclude that
LCM (14, 21) = 42

Question 7.
LCM (8, 25) = _______
We know that,
“LCM” is the lowest common divisor of the 2 numbers that are evenly divisible by  both the numbers
Now,
The multiples of 8 are:
8 = 8, 16, 24, 32 …………. 200
The multiples of 25 are:
25 = 25, 50, 75, 100 …….. 200
Hence, from the above,
We can conclude that
LCM (8, 25) = 200

Marathon Mathematics
In 2006, Deena Kastpor set the U.S. women’s record for both the half marathon (13.1 miles) and the full marathon (26.2 miles). Her time for the half marathon was 1 hour 7 minutes 34 seconds. Her time for the full marathon was 2 hours 19 minutes 36 seconds.
Question 1.
Compare her rates (seconds per mile) for the two races.
a. Which rate was faster?
It is given that
Deena Kastpor set the U.S. women’s record for both the half marathon (13.1 miles) and the full marathon (26.2 miles). Her time for the half marathon was 1 hour 7 minutes 34 seconds. Her time for the full marathon was 2 hours 19 minutes 36 seconds.
Now,
We have to convert the given time into seconds
We know that,
1 hour = 3600 seconds
1 minute = 60 seconds
So,
1 hour 7 minutes and 34 seconds = 1 (3600 seconds) + 7 (60 seconds) + 34 seconds
= 3600 seconds + 420 seconds + 34 seconds
= 4,054 seconds
2 hours 19 minutes and 36 seconds = 2 (3600 seconds) + 19 (60 seconds) + 36 seconds
= 7,200 seconds + 1140 seconds + 36 seconds
= 8,376 seconds
The rate at which half marathon completed (Seconds per Mile) = $$\frac{The time taken to complete half marathon in seconds}{The distance record for half marathon in miles}$$
= $$\frac{4,054}{13.1}$$
= 309 miles per second
The rate at which full marathon completed (Seconds per Mile) = $$\frac{The time taken to complete a full marathon in seconds}{The distance record for a full marathon in miles}$$
= $$\frac{8,376}{26.2}$$
= 319 miles per second
Hence, from the above,
We can conclude that the rate is faster for the “half marathon”

b. How much faster is her rate for that race?
From part (a),
The rate at which the half marathon completed is: 309 miles per second
The rate at which the full marathon completed is: 319 miles per second
So,
The amount of faster rate = 319 – 309
= 10 seconds per mile
Hence, from the above,
We can conclude that she ran about 10 seconds per mile faster in the half marathon

Question 2.
If Deena could run a full marathon at her half-marathon pace, how long would it take her to run the full marathon?

Question 3.
Which record do you think would be easier to break: half marathon or full marathon? Explain.
The half marathon record would be easier to break because it is more likely that you could increase speed for a short distance

Practice
Find the value of x that makes each number sentence true.
Question 4.
6x = 54 _______
The given expression is:
6x = 54
Divide by 6 on both sides
So,
$$\frac{6x}{6}$$ = $$\frac{54}{6}$$
x = 9
Hence, from the above,
We can conclude that the value of x for the given expresion is: 9

Question 5.
x – 14 = 152 _____
The given expression is:
x – 14 = 152
Add with 14 on both sides
So,
x – 14 + 14 = 152 + 14
x = 166
Hence, from the above,
We can conclude that the value of x for the given expression is: 166

Question 6.
300 = x + 199 ________
The given expression is:
300 = x + 199
Subtract with 199 on both sides
So,
300 – 199 = x + 199 – 199
x = 101
Hnece, from the above,
We can conclude that the value of x for the given expression is: 101

Doing the Dishes
Question 1.
Ronald’s family washes dishes by hand. Hand washing the dinner dishes takes about 10 minutes, and the faucet is running the whole time. The kitchen faucet runs at about 2.2 gallons per minute.
a. In one evening, about how much water does Ronald’s family use to wash dinner dishes?
It is given that
Ronald’s family washes dishes by hand. Hand washing the dinner dishes takes about 10 minutes, and the faucet is running the whole time. The kitchen faucet runs at about 2.2 gallons per minute.
So,
In one evening,
The dinner dishes washing only takes about 10 minutes
So,
The amount of water Ronald’s family used to wash dishes in one evening = (The time taken for hand washing the dishes by Ronald’s family) × ( The rate at which the kitchen faucet runs)
= 10 minutes × 2.2 gallons per minute
= 22 gallons
Hence, from the above,
We can conclude that
The amount of water used by Ronald’s family to wash dishes in one evening is: 22 gallons

b. In seven evenings, how much water does the family use to wash dishes?
From part (a),
We know that,
The amount of water used by Ronald’s family to wash dishes in 1 evening is: 22 gallons
So,
The amount of water used by Ronald’s family to wash dishes in seven evenings) = ( The amount of water used by Ronald’s family to wash dishes in 1 evening) × 7
= 22 × 7
= 154 gallons
Hence, from the above,
We can conclude that
The amount of water used by Ronald’s family to wash dishes in seven evenings is: 154 gallons

Question 2.
A high-efficiency faucet runs at about 1.5 gallons per minute.
a. About how much water would the family save each time they wash their dinner dishes if they replace their old faucet with a high-efficiency faucet?
The amount of water Ronald’s family used to wash dishes in one evening by using a high-efficiency faucet = (The time taken for hand washing the dishes by Ronald’s family) × ( The rate at which the  high-efficiency kitchen faucet runs)
= 10 minutes × 1.5 gallons per minute
= 15 gallons
Now,
The amount of water Ronald’s family used to wash dishes in one evening = (The time taken for hand washing the dishes by Ronald’s family) × ( The rate at which the kitchen faucet runs)
= 10 minutes × 2.2 gallons per minute
= 22 gallons
So,
The amount of water the family would save each time they wash their dinner dishes = 22 – 15
= 7 gallons
Hence, from the above,
We can conclude that
The amount of water the family would save each time they wash their dishes is: 7 gallons

b. About how much water would they save washing dinner dishes in a year (365 days)?
From part (a),
We know that,
The amount of water the family would save each time they wash their dishes is: 7 gallons
So,
The amount of water saved by the family they wash their dishes in 365 days
= 365 × 7
= 2.555 gallons
Hence, from the above,
We can conclude that
The amount of water saved by the family they wash their dishes in 365 days is: 2,555 gallons

Question 3.
A high-efficiency dishwasher uses about 4 gallons of water per load. The family would run the dishwasher 4 times per week to do their dinner dishes. Should they install a high-efficiency faucet (see Problem 2) or use the dishwasher to save water? Explain.
The amount of water Ronald’s family used to wash dishes in 7 days by using a high-efficiency faucet = (The time taken for hand washing the dishes by Ronald’s family) × ( The rate at which the  high-efficiency kitchen faucet runs)
= 10 minutes × 1.5 gallons per minute × 7
= 15 gallons × 7
= 105 gallons
The amount of water used by a high-efficiency dishwasher = (The amount of water used per load by a high-efficiency dishwasher) × ( The number of times the family would run dishwasher per week)
= 4 × 4
= 16 gallons
Hence, from the above,
We can conclude that we will use the dishwasher to save the water

Try This
Question 4.
A typical circular pool that is 18 feet across and 4 feet deep requires about 3,800 gallons of water. Ronald’s parents agree to get this pool if they cut their water usage enough to fill the pool. If they use the dishwasher, can Ronald’s family save enough water during the year to justify getting the pool? Explain.

Practice
Write whether each number sentence is true or false.
Question 5.
4 × 7 > 6 × 3 + 4 ______
The given sentence is:
4 × 7 > 6 × 3 + 4
28 > 18 + 4
28 > 22
Hence, from the above,
We can conclude that the given sentence is: True

Question 6.
15 + 9 ≤ 6 × 4 _______
The given sentence is:
15 + 9 ≤ 6 × 4
24 ≤ 24
Hence, from the above,
We can conclude that the given sentence is: True

Representing Patterns Home Link 7-8 in Different Ways
Use the pattern below to answer the questions.
Hint: The perimeter of one trapezoid is 5, not 4.

Question 1.
In the space below, sketch and label Step 4 and Step 5 of the sequence
The given sequence is:

Hence,
Step – 4 and step – 5 of the given sequence is:

Question 2.
Complete the table, and record an equation to represent the rule for finding the perimeter.
Rule: __________

Rule:
The perimeter can be found by using:
The perimeter of Step = Step n – 1 × The step number
We know that,
The perimeter of a trapezium = The sum of all of the sides of a trapezium
So,
The perimeter of a trapezium in Step 1 = 1 + 1 + 1 + 2
= 5 units
The perimeter of the trapezium in step 2 = (The perimeter of the trapezium in step 1) × (The step number)
= 5 units × 2
= 10 units
Hence,
The completed table of the perimeter of the trapezium is:

Question 3.
Use the values in the table from Problem 2 as the x- and y-coordinates for points. Graph the points on the coordinate grid.

From problem 2,

Now,
The ordered pairs for the graph o the coordinate grid are:
(1, 5), (2, 10), (3, 15), (4, 20), (5, 25), (10, 50)
Hence,
The graph of the ordered pairs in the coordinate grid is:

Practice
Evaluate.
Question 4.
15% of 80 = ______
The given expression is:
15% of 80
So,
15% of 80
= $$\frac{15}{100}$$ × 80
= 0.15 × 80
= 12
Hence, from the above,
We can conclude that the value of the given expression is: 12

Question 5.
45% of 200 = _______
The given expression is:
45% of 200
So,
45% of 200
= $$\frac{45}{100}$$ × 200
= 90
Hence, from the above,
We can conclude that the value of the given expression is: 90

Question 6.
85% of 2,200 = ________
The given expression is:
85% of 2,200
So,
85% of 2,200
= $$\frac{85}{100}$$ × 2,200
= 0.85 × 2,200
= 1,870
Hence, from the above,
We can conclude that the value of the given expression is: 1,870

Maximum Heart Rate
One way you can tell whether you are exercising too much, too little, or just the right amount is to check your heart rate. Calculate the number of beats per minute. The ideal average maximum heart rate is calculated by subtracting your age from 220.

Question 1.
Write an equation that represents the rule for calculating your ideal maximum heart rate. Rule: _____________
It is given that
The ideal average maximum heart rate is calculated by subtracting your age from 220
Now,
Let the age be x
Let the ideal maximum heart rate be y
So,
y = 220 – x
Hence, from the above,
We can conclude that the equation that represents the rule for calculating your ideal maximum heart rate is:
y = 220 – x

Question 2.
Use your rule to complete the table at the right with the beats per minute.

From Question 1,
The equation that represents the rule for calculating your ideal maximum heart rate is:
y = 220 – x
Hence,
The completed table is:

Question 3.
Explain how you know which variable is independent and which is dependent.
From Question 1,
The equation that represents the rule for calculating your ideal maximum heart rate is:
y = 220 – x
So,
From the equation,
We can observe that the maximum ideal heart rate is dependent on age and the age is independent
Hence, from the above,
We can conclude that
Independent variable: Age (x)
Dependent variable: Maximum ideal heart rate (y)

Question 4.
Graph the values in the table from Problem 2 as the x- and y-coordinates for points.

From problem 2,
The completed table is:

So,
The ordered pairs for the graph is:
(5, 215), (12, 208), (20, 200), (45, 175), (60, 160)
Hence,
The representation of the ordered pairs in the graph is:

Practice
Evaluate.
Question 5.
-(4) = _______
We know that,
– * + = –
+ * + = +
+ * – = –
– * – = +
So,
-(4) = – (+4)
= -4

Question 6.
-(-9) = ______
We know that,
– * + = –
+ * + = +
+ * – = –
– * – = +
So,
– (-9) = +9
= 9

Question 7.
-(-1.5) = ______
We know that,
– * + = –
+ * + = +
+ * – = –
– * – = +
So,
– (-1.5) = +1.5
= 1.5

Comparing Tables Home Link 7-10 and Graphs
Question 1.
Complete the tables for squares with the given side lengths

We know that,
The perimeter of the square = 4 × Side Length
The area of the square = Side²
Hence,
The completed tables of perimeters and areas for squares with the given side lengths is:

Question 2.
Use the values in the tables in Problem 1 to make graphs for perimeter and area.

From problem 1,
The completed table of the perimeter and area for the squares with the given side lengths is:

Now,
For the graph of the perimeter of the square,
The ordered pairs are:
(1, 4), (2, 8), (3, 12), (4, 16), (5, 20)
For the graph of the area of the square,
The ordered pairs are:
(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)
Hence,
The graphs for the perimeter and area of the square are:
The perimeter of the square:

The area of the square:

Question 3.
Explain why the graphs look different.
The area grows faster when compared to the perimeter
The change in perimeter remains constant whereas the change in the area changes every time
Hence,
The graphs of the perimeter and the area look different

Practice
Find each number based on the given percents.
Question 4.
10% of n is 4; n = ________
It is given that
10% of n is 4
So,
10% of n = 4
We know that,
1% = 0.01
So,
10% = 0.10
So,
0.10 of n = 4
0.10 × n = 4
Divide by 0.10 on both sides
So,
$$\frac{0.10}{0.10}$$ × n = $$\frac{4}{0.10}$$
n = 4 ×10
n = 40
Hence, from the above,
We can conclude that the value of n is: 40

Question 5.
30% of n is 18; n = __________
It is given that
30% of n is 18
So,
30% of n = 18
We know that,
1% = 0.01
So,
30% = 0.30
So,
0.30 of n = 18
0.30 × n = 18
Divide by 0.30 on both sides
So,
$$\frac{0.30}{0.30}$$ × n = $$\frac{18}{0.30}$$
n = 6 ×10
n = 60
Hence, from the above,
We can conclude that the value of n is: 60

Mystery Graphs
Question 1.
Create a mystery graph on the grid below. Be sure to label the horizontal and vertical axes. Describe the situation that goes with your graph on the lines provided.

For the mystery graph,
Let the horizontal axis be: x
Let the vertical axis be: y
Let the ordered pairs for the mystery graph be:
(1, 2), (2, 5), (3, 7), (4, 9), (5, 12), (6, 15), (7, 18), (8, 22), (9, 25), (10, 30)
Hence,
The representation of the mystery graph with the above-ordered pairs is:

Now,
From the above graph,
The situation is:
When the value of x increases, the value of y also increases and the rate of change is not constant
So,
The shape of the mystery graph looks like a curve

Practice
Compute. Use the back of the page to do the computation.
Question 2.
432 ÷ 3
The given expression is:
432 ÷ 3
So,

Hence, from the above,
We can conclude that
432 ÷ 3 = 144

Question 3.
2,412 ÷ 2
The given expression is:
2,412 ÷ 2
So,

Hence, from the above,
We can conclude that
2,412 ÷ 2 = 201

Question 4.
1,325 ÷ 5