Real Equations Word Problems Worksheet
Algebra 1 • Real Numbers CCSS 8.EE.C.7 • HSA-CED.A.1
Real World Equations
Formulate a real number equation for each situation, solve with clear steps, and label units.
Name:
Date:
Period: Score: / 10
Solving Steps:
1. Define variable 2. Set up equation 3. Use inverse operations 4. State answer with units
01
Mobile Data: Maya's phone plan charges a flat $24.50 monthly fee plus $8.75 for each gigabyte (\(g\)) of high-speed data over her limit. If her total bill was $68.25, write and solve an equation to find how many extra gigabytes she used.
Define & Equation
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Solution
02
Woodworking Cut: A carpenter has an oak board \(9\frac{1}{2}\) feet long. He cuts off \(3\frac{1}{4}\) feet for a shelf bracket and divides the remaining length into 5 equal support struts of length \(s\). Write and solve an equation to find the length in feet of each strut.
Define & Equation
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Solution
03
Alpine Temperature: At dawn, an alpine sensor measured \(-14.2^\circ\text{C}\). Over the next 6 hours, the temperature rose at a constant hourly rate of \(r\) degrees until reaching \(3.8^\circ\text{C}\). Write and solve an equation to find the hourly rate of change \(r\).
Define & Equation
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Solution
04
Studio Rental: Soundstage A charges a $35.00 setup fee plus $22.50 per hour. Soundstage B charges a $12.50 setup fee plus $27.00 per hour. Write and solve an equation to find the number of hours \(h\) for which both studios charge the exact same total.
Define & Equation
Show Work
Solution
05
Retail Sale & Voucher: Leo bought a drawing tablet at 35% off its sticker price \(p\). He also used a $15.00 member voucher on the discounted price. If Leo's pre-tax total was $147.50, write and solve an equation to find the original sticker price \(p\).
Define & Equation
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Solution
REAL WORLD EQUATIONS • WORKBOOK SET PAGE 1 OF 2
Section 2 Real World Equations Practice
Student Name:
06
Garden Perimeter: A rectangular community garden has a perimeter of \(43\frac{1}{2}\) feet and a width of \(8\frac{1}{4}\) feet. If the length is \(\ell\), write and solve an equation using \(P = 2\ell + 2w\) to find the length in feet of the garden.
Define & Equation
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Solution
07
Bicycle Savings Goal: Jasmine has $185.40 saved toward a $520.00 mountain bicycle. Each week, she deposits $27.80 from tutoring and $14.10 from allowance. Write and solve an equation to find how many weeks \(w\) it will take to reach her goal.
Define & Equation
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Solution
08
Submersible Descent: A research sub begins at an elevation of \(-85.5\) meters relative to sea level. It descends at a rate of \(12.4\) meters per minute for \(m\) minutes until reaching \(-234.3\) meters. Write and solve an equation to find \(m\).
Define & Equation
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Solution
09
Bakery Inventory: A baker uses \(\frac{3}{8}\) pound of butter for each brioche loaf (\(b\)), plus \(1\frac{1}{2}\) pounds of butter for the morning glaze batch. If \(18\) pounds of butter were used in total, write and solve an equation to find the loaves made.
Define & Equation
Show Work
Solution
10
Group Passes: A coach buys 12 adventure park passes. With a promo code, each pass received a $4.50 discount off the standard adult rate \(r\). After a flat $15.00 service fee, the total was $285.00. Write and solve an equation to determine \(r\).
Define & Equation
Show Work
Solution
REAL WORLD EQUATIONS • WORKBOOK SET PAGE 2 OF 2
Real Equations Answer Key
Teacher Guide • Complete Key Standards 8.EE.C.7 • HSA-CED.A.1
Real World Equations — Answer Key
Model formulations, algebraic steps with inverse operations, checks, and common pitfalls.
Grading Scale (30 pts)
1 pt: Correct Equation Model
1 pt: Complete Inverse Ops
1 pt: Accurate Units & Value
01 Mobile Data Overage
g = 5 gigabytes
Model Setup
Let \(g\) = extra GB used
24.50 + 8.75g = 68.25
Algebraic Solution
8.75g = 68.25 - 24.50
8.75g = 43.75
g = 43.75 / 8.75 ⇒ g = 5
Check & Teaching Tip
\(24.50 + 8.75(5) = 24.50 + 43.75 = 68.25\) ✓
Watch for: subtracting $8.75 from $68.25 instead of the $24.50 base charge.
02 Woodworking Board Division
s = 1¼ ft (1.25 ft)
Model Setup
Let \(s\) = strut length (ft)
3¼ + 5s = 9½ or 3.25 + 5s = 9.5
Algebraic Solution
5s = 9½ - 3¼ = 6¼ (or 6.25)
5s = 25/4 ⇒ s = (25/4) × (1/5)
s = 5/4 = 1¼ ft (or 1.25 ft)
Check & Teaching Tip
\(3.25 + 5(1.25) = 3.25 + 6.25 = 9.5\) ✓
Tip: Converting to improper fractions or terminating decimals both yield clean work.
03 Alpine Temperature Rise
r = 3.0°C / hour
Model Setup
Let \(r\) = rate of rise (°C/hr)
-14.2 + 6r = 3.8
Algebraic Solution
6r = 3.8 - (-14.2)
6r = 3.8 + 14.2 = 18.0
r = 18.0 / 6 ⇒ r = 3.0°C/hr
Check & Teaching Tip
\(-14.2 + 6(3) = -14.2 + 18 = 3.8\) ✓
Common error: Calculating \(3.8 - 14.2 = -10.4\) instead of adding 14.2.
04 Soundstage Rental Break-Even
h = 5 hours ($147.50)
Model Setup
Let \(h\) = rental hours
35 + 22.50h = 12.50 + 27.00h
Algebraic Solution
35 - 12.50 = 27.00h - 22.50h
22.50 = 4.50h
h = 22.50 / 4.50 ⇒ h = 5 hours
Check & Teaching Tip
Studio A: \(35 + 22.5(5) = \$147.50\)
Studio B: \(12.5 + 27(5) = \$147.50\) ✓
Concept: Variables on both sides; keep variable coefficient positive.
05 Retail Sale & Voucher
p = $250.00
Model Setup
Let \(p\) = original sticker price
35% off ⇒ pays 65% (0.65)
0.65p - 15 = 147.50
Algebraic Solution
0.65p = 147.50 + 15
0.65p = 162.50
p = 162.50 / 0.65 ⇒ p = $250.00
Check & Teaching Tip
\(0.65(250) - 15 = 162.50 - 15 = 147.50\) ✓
Common error: Using \(0.35p\) instead of \((1 - 0.35)p = 0.65p\).
REAL WORLD EQUATIONS • TEACHER ANSWER KEY PAGE 1 OF 2
Teacher Key • Section 2 Problems 6 to 10 Solutions
Answer Key • Verification Completed
06 Garden Perimeter Geometry
ℓ = 13½ ft (13.5 ft)
Model Setup
Let \(\ell\) = length in feet
2ℓ + 2(8¼) = 43½
2ℓ + 16.5 = 43.5
Algebraic Solution
2ℓ = 43.5 - 16.5
2ℓ = 27.0
ℓ = 27.0 / 2 ⇒ ℓ = 13.5 ft (13½ ft)
Check & Teaching Tip
\(2(13.5) + 2(8.25) = 27 + 16.5 = 43.5\) ✓
Alternative: \(2(\ell + 8.25) = 43.5 \implies \ell + 8.25 = 21.75 \implies \ell = 13.5\).
07 Bicycle Savings Goal
w = 8 weeks
Model Setup
Let \(w\) = weeks of savings
Weekly: $27.80 + $14.10 = $41.90
185.40 + 41.90w = 520.00
Algebraic Solution
41.90w = 520.00 - 185.40
41.90w = 334.60
w = 334.60 / 41.90 ⇒ w = 8 weeks
Check & Teaching Tip
\(185.40 + 41.90(8) = 185.40 + 334.60 = 520.00\) ✓
Key step: Combining like terms (tutoring + allowance) before isolating.
08 Submersible Descent (Signed Real Numbers)
m = 12 minutes
Model Setup
Let \(m\) = descent minutes
-85.5 - 12.4m = -234.3
Algebraic Solution
-12.4m = -234.3 - (-85.5)
-12.4m = -234.3 + 85.5 = -148.8
m = -148.8 / -12.4 ⇒ m = 12 minutes
Check & Teaching Tip
\(-85.5 - 12.4(12) = -85.5 - 148.8 = -234.3\) ✓
Sign awareness: Dividing a negative by a negative yields positive time.
09 Bakery Brioche Loaves
b = 44 loaves
Model Setup
Let \(b\) = number of brioche loaves
⅜b + 1½ = 18
0.375b + 1.5 = 18
Algebraic Solution
⅜b = 18 - 1½ = 16½ (33/2)
b = (33/2) × (8/3)
b = (33 × 8) / (2 × 3) = 264 / 6 ⇒ b = 44
Check & Teaching Tip
\(\frac{3}{8}(44) + 1.5 = 16.5 + 1.5 = 18\) ✓
Fraction skill: Multiplying by reciprocal \(\frac{8}{3}\) avoids division errors.
10 Adventure Park Group Passes (Distributive Property)
r = $27.00 per pass
Model Setup
Let \(r\) = standard adult rate
12(r - 4.50) + 15 = 285
or 12r - 54 + 15 = 285
Algebraic Solution
Method A: 12(r - 4.50) = 270
r - 4.50 = 22.50 ⇒ r = $27.00
Method B: 12r - 39 = 285 ⇒ 12r = 324 ⇒ r = 27
Check & Teaching Tip
\(12(27 - 4.50) + 15 = 12(22.50) + 15 = 270 + 15 = 285\) ✓
Distributive trap: Forgetting to multiply \(12 \times -4.50 = -54\).
Teacher Diagnostic & Error Analysis Matrix
Rational Numbers (Fractions)
Problems 2 & 9: If students struggle with \(\frac{3}{8}\) or mixed numbers, remind them to clear denominators by multiplying both sides by the LCD.
Signed Decimals (Negatives)
Problems 3 & 8: Reinforce that subtracting a negative value is equivalent to adding its positive opposite (\(3.8 - (-14.2) = 18.0\)).
Multi-Step Distribution
Problems 4, 5 & 10: Watch for discount rate conversion (paying 65% for 35% off) and distributing factors to all terms inside parentheses.
REAL WORLD EQUATIONS • TEACHER ANSWER KEY PAGE 2 OF 2
