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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 1

Write Equations

Turn a sentence into an equation in one variable.

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Key Ideas Easy Medium Rate of Change Medium Perimeter Hard Consecutive Integers Super Challenge
What to Remember

Name the variable first

Say what the letter stands for before you write the math. Include units when the story has them (hours, dollars, tickets, gallons).

Translate the words

is / equals → =. Write the same value on both sides.

sum, increased by, more than, plus → add. difference, decreased by, less than, minus → subtract.

product, times, of, twice → multiply. quotient, divided by, half of → divide.

squared is n². cubed is n³.

Order matters when you subtract

“5 less than n” is n − 5, not 5 − n.

“the difference of n and 5” is n − 5. “the difference of 5 and n” is 5 − n.

Starting amount and a rate

Increasing: (amount) = (start) + (rate)(quantity).

Decreasing: (amount) = (start) − (rate)(quantity). The rate is the number that multiplies the variable.

Perimeter of a rectangle

Write one side in terms of the other, then use P = 2ℓ + 2w.

Substitute the expressions for the sides. Do not plug in a guessed number for ℓ or w.

Consecutive integers

Consecutive integers differ by 1: n and n + 1. Three in a row: n, n + 1, n + 2.

Consecutive even integers and consecutive odd integers both differ by 2: n and n + 2. Three in a row: n, n + 2, n + 4.

Level 1

Easy

One sentence. Name the variable, then write the equation.

Example
Write an equation. 9 more than a number n is 24.

“9 more than n” is n + 9.

is means =.

n + 9 = 24

Answer: n + 9 = 24

1
Write an equation. The sum of a number n and 17 is 41.
Show Answer

The sum of n and 17 is n + 17.

n + 17 = 41

Answer: n + 17 = 41

2
Write an equation. The difference of a number n and 38 is 11.
Show Answer

The difference of n and 38 is n − 38.

n − 38 = 11

Answer: n − 38 = 11

3
Write an equation. The quotient of a number n and 7 is 8.
Show Answer

The quotient of n and 7 is n7.

n7 = 8

Answer: n7 = 8

4
Write an equation. A number n decreased by 15 is 22.
Show Answer

n decreased by 15 is n − 15.

n − 15 = 22

Answer: n − 15 = 22

5
Write an equation. Half of a number n is 16.
Show Answer

Half of n is n2.

n2 = 16

Answer: n2 = 16

6
Write an equation. A number n increased by 9 is 31.
Show Answer

n increased by 9 is n + 9.

n + 9 = 31

Answer: n + 9 = 31

7
Write an equation. The product of a number n and 9 is 72.
Show Answer

The product of n and 9 is 9n.

9n = 72

Answer: 9n = 72

8
Write an equation. A number n squared is 81.
Show Answer

n squared is n².

n² = 81

Answer: n² = 81

9
Write an equation. Twice a number n is 28.
Show Answer

Twice n is 2n.

2n = 28

Answer: 2n = 28

10
Write an equation. The quotient of 42 and a number n is 6.
Show Answer

The quotient of 42 and n is 42n.

42n = 6

Answer: 42n = 6

11
Write an equation. A number n cubed is 64.
Show Answer

n cubed is n³.

n³ = 64

Answer: n³ = 64

12
Write an equation. If 8 is subtracted from a number n, the result is 19.
Show Answer

8 subtracted from n is n − 8.

n − 8 = 19

Answer: n − 8 = 19

13
Write an equation. If four times a number n is increased by 7, the result is 35.
Show Answer

Four times n is 4n.

Increased by 7 is 4n + 7.

4n + 7 = 35

Answer: 4n + 7 = 35

14
Write an equation. When 9 is subtracted from five times a number n, the result is 16.
Show Answer

Five times n is 5n.

9 subtracted from that is 5n − 9.

5n − 9 = 16

Answer: 5n − 9 = 16

15
Write an equation. The sum of 10 and 5 is equal to the difference of a number n and 8.
Show Answer

The sum of 10 and 5 is 10 + 5.

The difference of n and 8 is n − 8.

10 + 5 = n − 8

Answer: 10 + 5 = n − 8

16
Write an equation. The sum of three times a number n and 6 is 27.
Show Answer

Three times n is 3n.

The sum of that and 6 is 3n + 6.

3n + 6 = 27

Answer: 3n + 6 = 27

17
Write an equation. The sum of twice a number n and 11 is 29.
Show Answer

Twice n is 2n.

The sum of that and 11 is 2n + 11.

2n + 11 = 29

Answer: 2n + 11 = 29

18
Write an equation. Seven times a number n decreased by 4 is 38.
Show Answer

Seven times n is 7n.

Decreased by 4 is 7n − 4.

7n − 4 = 38

Answer: 7n − 4 = 38

19
Write an equation. If 5 is added to the sum of twice a number n and three times the number, the result is 35.
Show Answer

Twice n is 2n. Three times n is 3n.

Their sum is 2n + 3n.

5 added to that is 5 + (2n + 3n).

5 + (2n + 3n) = 35

Answer: 5 + (2n + 3n) = 35

20
Write an equation. 8 less than 3 times a number n is 19.
Show Answer

3 times n is 3n.

8 less than that is 3n − 8.

3n − 8 = 19

Answer: 3n − 8 = 19

21
Write an equation. Three fifths of a number n is −9.
Show Answer

of means multiply.

35n = −9

Answer: 35n = −9

22
Write an equation. A number n is two tenths less than 3.5.
Show Answer

two tenths less than 3.5 is 3.5 − 210.

n = 3.5 − 210

Answer: n = 3.5 − 210

23
Write an equation. Four tenths of a number n is 1.6.
Show Answer

of means multiply.

410n = 1.6

Answer: 410n = 1.6

24
Write an equation. The difference of 2 times a number n and 13 is 56.
Show Answer

2 times n is 2n.

The difference of 2n and 13 is 2n − 13.

2n − 13 = 56

Answer: 2n − 13 = 56

25
Write an equation. The product of a number n and 1.5 is 6.
Show Answer

The product of n and 1.5 is 1.5n.

1.5n = 6

Answer: 1.5n = 6

26
Write an equation. Three eighths added to a number n is 2.
Show Answer

38 added to n is n + 38.

n + 38 = 2

Answer: n + 38 = 2

27
Write an equation. The product of 3.2 and 4.5 is equal to a number n times 0.8.
Show Answer

The product of 3.2 and 4.5 is 3.2 × 4.5.

n times 0.8 is 0.8n.

3.2 × 4.5 = 0.8n

Answer: 3.2 × 4.5 = 0.8n

Level 2

Medium Rate of Change

Linear situations. The amount can increase or decrease at a constant rate.

Example
A paddleboard rental costs $20 plus $8 per hour. Write an equation for the total cost C after h hours.

Let h = the number of hours.

Let C = the total cost in dollars.

Start plus rate times hours.

C = 20 + 8h

Answer: C = 20 + 8h

1
A scooter rental costs $9 plus $4 per hour. Write an equation for the total cost C after h hours.
Show Answer

Let h = the number of hours.

Let C = the total cost in dollars.

C = 9 + 4h

Answer: C = 9 + 4h

2
Maya has $125 in savings and adds $15 each week. Write an equation for the amount S after w weeks.
Show Answer

Let w = the number of weeks.

Let S = the amount in dollars.

S = 125 + 15w

Answer: S = 125 + 15w

3
Jordan has already hiked 4 miles and then walks at 3 miles per hour for t hours. Write an equation for the total distance D.
Show Answer

Let t = the number of hours.

Let D = the total distance in miles.

D = 4 + 3t

Answer: D = 4 + 3t

4
A pool starts with 80 gallons and fills at 6 gallons per minute. Write an equation for the amount A after m minutes.
Show Answer

Let m = the number of minutes.

Let A = the amount of water in gallons.

A = 80 + 6m

Answer: A = 80 + 6m

5
A printer has already printed 25 pages and then prints 12 pages per minute. Write an equation for the total pages P after t minutes.
Show Answer

Let t = the number of minutes.

Let P = the total number of pages.

P = 25 + 12t

Answer: P = 25 + 12t

6
A taxi charges $2.75 plus $0.90 per mile. Write an equation for the cost C after m miles.
Show Answer

Let m = the number of miles.

Let C = the cost in dollars.

C = 2.75 + 0.90m

Answer: C = 2.75 + 0.90m

7
A tank starts with 64 gallons and drains at 5 gallons per minute. Write an equation for the amount A after t minutes.
Show Answer

Let t = the number of minutes.

Let A = the amount of water in gallons.

A = 64 − 5t

Answer: A = 64 − 5t

8
A candle starts 12 inches tall and burns 0.6 inches per hour. Write an equation for the height H after h hours.
Show Answer

Let h = the number of hours.

Let H = the height in inches.

H = 12 − 0.6h

Answer: H = 12 − 0.6h

9
An account has $420, and $24 is spent each day. Write an equation for the balance B after d days.
Show Answer

Let d = the number of days.

Let B = the balance in dollars.

B = 420 − 24d

Answer: B = 420 − 24d

10
A balloon is at 1800 feet and descends 60 feet per minute. Write an equation for the height H after t minutes.
Show Answer

Let t = the number of minutes.

Let H = the height in feet.

H = 1800 − 60t

Answer: H = 1800 − 60t

Level 2

Medium Perimeter

Write a side in terms of the other, then use the perimeter formula.

Example
The length of a rectangle is 5 cm more than the width w. The perimeter is 38 cm. Write an equation for w.

Length = w + 5

Perimeter formula.

P = 2ℓ + 2w

Substitute.

38 = 2(w + 5) + 2w

Answer: 38 = 2(w + 5) + 2w

1
The width of a rectangle is 6 cm less than the length ℓ. The perimeter is 56 cm. Write an equation for ℓ.
Show Answer

Width = ℓ − 6

Perimeter formula.

P = 2ℓ + 2w

Substitute.

56 = 2ℓ + 2(ℓ − 6)

Answer: 56 = 2ℓ + 2(ℓ − 6)

2
The length of a rectangle is 4 cm more than twice the width w. The perimeter is 62 cm. Write an equation for w.
Show Answer

Length = 2w + 4

Perimeter formula.

P = 2ℓ + 2w

Substitute.

62 = 2(2w + 4) + 2w

Answer: 62 = 2(2w + 4) + 2w

3
The length of a rectangle is 3 times the width w. The perimeter is 48 cm. Write an equation for w.
Show Answer

Length = 3w

Perimeter formula.

P = 2ℓ + 2w

Substitute.

48 = 2(3w) + 2w

Answer: 48 = 2(3w) + 2w

4
The length of a rectangle is 7 cm more than the width w. The perimeter is 50 cm. Write an equation for w.
Show Answer

Length = w + 7

Perimeter formula.

P = 2ℓ + 2w

Substitute.

50 = 2(w + 7) + 2w

Answer: 50 = 2(w + 7) + 2w

5
The width of a rectangle is 2 cm more than the length ℓ. The perimeter is 40 cm. Write an equation for ℓ.
Show Answer

Width = ℓ + 2

Perimeter formula.

P = 2ℓ + 2w

Substitute.

40 = 2ℓ + 2(ℓ + 2)

Answer: 40 = 2ℓ + 2(ℓ + 2)

Level 3

Hard Consecutive Integers

Consecutive integers differ by 1. Consecutive even integers and consecutive odd integers differ by 2.

Example
The sum of two consecutive integers is 39. Let n be the smaller integer. Write an equation.

Consecutive integers differ by 1.

The integers are n and n + 1.

n + (n + 1) = 39

Answer: n + (n + 1) = 39

1
The sum of two consecutive integers is 61. Let n be the smaller integer. Write an equation.
Show Answer

Consecutive integers differ by 1.

The integers are n and n + 1.

n + (n + 1) = 61

Answer: n + (n + 1) = 61

2
The sum of three consecutive integers is 96. Let n be the smallest integer. Write an equation.
Show Answer

The integers are n, n + 1, and n + 2.

n + (n + 1) + (n + 2) = 96

Answer: n + (n + 1) + (n + 2) = 96

3
The sum of two consecutive even integers is 94. Let n be the smaller even integer. Write an equation.
Show Answer

Consecutive even integers differ by 2.

The integers are n and n + 2.

n + (n + 2) = 94

Answer: n + (n + 2) = 94

4
The sum of two consecutive odd integers is 48. Let n be the smaller odd integer. Write an equation.
Show Answer

Consecutive odd integers differ by 2.

The integers are n and n + 2.

n + (n + 2) = 48

Answer: n + (n + 2) = 48

5
The sum of three consecutive even integers is 84. Let n be the smallest even integer. Write an equation.
Show Answer

Consecutive even integers differ by 2.

The integers are n, n + 2, and n + 4.

n + (n + 2) + (n + 4) = 84

Answer: n + (n + 2) + (n + 4) = 84

6
The sum of three consecutive odd integers is 69. Let n be the smallest odd integer. Write an equation.
Show Answer

Consecutive odd integers differ by 2.

The integers are n, n + 2, and n + 4.

n + (n + 2) + (n + 4) = 69

Answer: n + (n + 2) + (n + 4) = 69

7
The sum of two consecutive integers is 55. Let n be the larger integer. Write an equation.
Show Answer

Let n = the larger integer.

The smaller integer is n − 1.

(n − 1) + n = 55

Answer: (n − 1) + n = 55

Super Challenge

Super Challenge

1
A repair costs $100 for the first 4 hours and $20 for each hour after that. The total cost is $180. Let t be the number of hours from the start. Write an equation for this situation after the first 4 hours.
Show Answer

Let t = the number of hours from the start.

Total cost = (cost of the first 4 hours) + (new rate)(hours after 4).

The first 4 hours cost $100. After that, the rate is $20 per hour.

t starts at the beginning, so the hours at $20 are t − 4.

100 + 20(t − 4) = 180

Answer: 100 + 20(t − 4) = 180

2
A parking garage charges $18 for the first 3 hours and $6 for each hour after that. The total cost is $54. Let t be the number of hours from the start. Write an equation for this situation after the first 3 hours.
Show Answer

Let t = the number of hours from the start.

Total cost = (cost of the first 3 hours) + (new rate)(hours after 3).

The first 3 hours cost $18. After that, the rate is $6 per hour.

t starts at the beginning, so the hours at $6 are t − 3.

18 + 6(t − 3) = 54

Answer: 18 + 6(t − 3) = 54

3
A kayak rental costs $45 for the first 2 hours and $12 for each hour after that. The total cost is $105. Let t be the number of hours from the start. Write an equation for this situation after the first 2 hours.
Show Answer

Let t = the number of hours from the start.

Total cost = (cost of the first 2 hours) + (new rate)(hours after 2).

The first 2 hours cost $45. After that, the rate is $12 per hour.

t starts at the beginning, so the hours at $12 are t − 2.

45 + 12(t − 2) = 105

Answer: 45 + 12(t − 2) = 105

4
A moving truck costs $80 for the first 5 hours and $25 for each hour after that. The total cost is $230. Let t be the number of hours from the start. Write an equation for this situation after the first 5 hours.
Show Answer

Let t = the number of hours from the start.

Total cost = (cost of the first 5 hours) + (new rate)(hours after 5).

The first 5 hours cost $80. After that, the rate is $25 per hour.

t starts at the beginning, so the hours at $25 are t − 5.

80 + 25(t − 5) = 230

Answer: 80 + 25(t − 5) = 230

5
A cabin rental costs $240 for the first 3 days and $70 for each day after that. The total cost is $590. Let t be the number of days from the start. Write an equation for this situation after the first 3 days.
Show Answer

Let t = the number of days from the start.

Total cost = (cost of the first 3 days) + (new rate)(days after 3).

The first 3 days cost $240. After that, the rate is $70 per day.

t starts at the beginning, so the days at $70 are t − 3.

240 + 70(t − 3) = 590

Answer: 240 + 70(t − 3) = 590

6
A local bakery delivered a total of 85 pastries to a morning conference. The delivery consisted of croissants, blueberry muffins, and cinnamon rolls. Let c be the number of croissants. There were twice as many blueberry muffins as croissants. There were 10 fewer cinnamon rolls than croissants. Write a simplified equation for the number of pastries delivered. Do not solve.
Show Answer

Let c = the number of croissants.

Blueberry muffins: 2c

Cinnamon rolls: c − 10

Croissants + blueberry muffins + cinnamon rolls = 85.

c + 2c + (c − 10) = 85

4c − 10 = 85

Answer: 4c − 10 = 85

7
A florist arranged a total of 92 stems for a wedding centerpiece. The arrangement consisted of roses, tulips, and daisies. Let r be the number of roses. There were 3 times as many tulips as roses. There were 8 fewer daisies than roses. Write a simplified equation for the number of stems arranged. Do not solve.
Show Answer

Let r = the number of roses.

Tulips: 3r

Daisies: r − 8

Roses + tulips + daisies = 92.

r + 3r + (r − 8) = 92

5r − 8 = 92

Answer: 5r − 8 = 92

8
An art teacher bought a bulk pack of 40 specialty drawing tools for a printmaking class. The pack consisted of palette knives, bamboo brushes, and detailing pens. Let p be the number of palette knives. The teacher bought 3 times as many 8-inch bamboo brushes as palette knives, and 5 more 4-inch detailing pens than palette knives. Write a simplified equation in terms of p. Do not solve.
Show Answer

Let p = the number of palette knives.

8-inch is the length of each bamboo brush. The number of brushes is 3p.

4-inch is the length of each detailing pen. The number of pens is p + 5.

Palette knives + bamboo brushes + detailing pens = 40.

p + 3p + (p + 5) = 40

5p + 5 = 40

Answer: 5p + 5 = 40

9
A coach set out 43 pieces of practice equipment for a drill. The equipment consisted of cones, agility ladders, and hurdles. Let n be the number of cones. The coach set out twice as many 15-foot agility ladders as cones, and 7 more 6-inch hurdles than cones. Write a simplified equation in terms of n. Do not solve.
Show Answer

Let n = the number of cones.

15-foot is the length of each agility ladder. The number of ladders is 2n.

6-inch is the height of each hurdle. The number of hurdles is n + 7.

Cones + agility ladders + hurdles = 43.

n + 2n + (n + 7) = 43

4n + 7 = 43

Answer: 4n + 7 = 43

10
An outdoor club packed 44 pieces of gear for a climbing day. The gear consisted of ropes, carabiners, and stakes. Let g be the number of carabiners. The club packed 5 times as many 50-foot ropes as carabiners, and 2 more 8-inch stakes than carabiners. Write a simplified equation in terms of g. Do not solve.
Show Answer

Let g = the number of carabiners.

50-foot is the length of each rope. The number of ropes is 5g.

8-inch is the length of each stake. The number of stakes is g + 2.

Ropes + carabiners + stakes = 44.

g + 5g + (g + 2) = 44

7g + 2 = 44

Answer: 7g + 2 = 44

11
In a typical week the textile conservation lab at the municipal museum treats 80 garments. One bottle of pH-neutral rinse makes enough solution to treat 16 garments. The lab wants to lower its weekly spending on the concentrate by $6. The concentrate costs $5 per bottle. Write an equation the lab can use to determine how many fewer garments, g, it should treat each week. Do not solve.
Show Answer

Let g = how many fewer garments to treat each week.

80 is the number of garments treated in a typical week. The equation uses the cost of one garment and the $6 savings.

One bottle costs $5 and treats 16 garments.

The cost of one garment is 516 dollars.

The cost of g fewer garments is the $6 the lab wants to save.

516g = 6

Answer: 516g = 6

12
A maple-sugaring co-op boils down an average of 90 gallons of sap each week. One cord of firewood runs the evaporator long enough to process 20 gallons of sap. The co-op wants to cut its weekly firewood bill by $5. Firewood costs $7 per cord. Write an equation the co-op can use to determine how many fewer gallons of sap, v, it should boil each week. Do not solve.
Show Answer

Let v = how many fewer gallons of sap to boil each week.

90 is the number of gallons boiled in a typical week. The equation uses the cost of one gallon and the $5 savings.

One cord costs $7 and processes 20 gallons.

The cost of one gallon is 720 dollars.

The cost of v fewer gallons is the $5 the co-op wants to save.

720v = 5

Answer: 720v = 5

13
A neon-sign workshop fills an average of 75 meters of glass tubing each week. One cylinder of noble gas services 18 meters of tubing. The shop wants to reduce its weekly spending on gas by $4. Cylinders cost $5 each. Write an equation the shop can use to determine how many fewer meters of tubing, t, it should fill each week. Do not solve.
Show Answer

Let t = how many fewer meters of tubing to fill each week.

75 is the number of meters filled in a typical week. The equation uses the cost of one meter and the $4 savings.

One cylinder costs $5 and services 18 meters.

The cost of one meter is 518 dollars.

The cost of t fewer meters is the $4 the shop wants to save.

518t = 4

Answer: 518t = 4

14
A community wood-fired bakery loads an average of 70 trays each week. One bundle of kiln-dried oak lasts for 14 trays. The bakery wants to lower its weekly spending on wood by $5. Bundles cost $3 each. Write an equation the bakery can use to determine how many fewer trays, w, it should load each week. Do not solve.
Show Answer

Let w = how many fewer trays to load each week.

70 is the number of trays loaded in a typical week. The equation uses the cost of one tray and the $5 savings.

One bundle costs $3 and lasts for 14 trays.

The cost of one tray is 314 dollars.

The cost of w fewer trays is the $5 the bakery wants to save.

314w = 5

Answer: 314w = 5

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