Algebra 1
10 Core Word Problems
Quick Memory Keys
RATE = distance ÷ time
TOTAL = unit × quantity
EQUAL → set expressions =
CONSECUTIVE → n, n+1, n+2
PERCENT → multiply by 0.xx
INVERSE → flip the operation
A — 01
Sarah has $240 and spends $15 each day. Marcus has $60 and earns $9 each day. After how many days will they have the same amount of money?
⚠️ Trick: Sarah's money decreases — use subtraction, not addition
Solution: Let d = number of days.
Since we need a whole day, answer is approximately 8 days. Key: one goes up, one goes down → opposite signs.
Sarah: 240 − 15d
Marcus: 60 + 9d
Set equal: 240 − 15d = 60 + 9d
180 = 24d → d = 7.5 days
Since we need a whole day, answer is approximately 8 days. Key: one goes up, one goes down → opposite signs.
A — 02
The sum of three consecutive odd integers is 57. What is the largest of the three integers?
⚠️ Trick: Consecutive odd integers differ by 2, not 1
Solution: Let integers = n, n+2, n+4
n + (n+2) + (n+4) = 57
3n + 6 = 57 → 3n = 51 → n = 17
Largest = n + 4 = 21
Memory key: odd/even consecutive → use n, n+2, n+4
A — 03
A train travels from City A to City B at 60 mph. The return trip is at 40 mph. If the total trip takes 5 hours, how far is it from A to B?
⚠️ Trick: Average speed ≠ (60+40)/2. Use total distance / total time.
Solution: Let d = one-way distance.
Time A→B: d/60 Time B→A: d/40
d/60 + d/40 = 5
LCD = 120: 2d/120 + 3d/120 = 5
5d/120 = 5 → d = 120 miles
A — 04
A store sells apples for $0.75 each and oranges for $1.25 each. Jake buys a total of 20 fruits and pays exactly $19. How many oranges did he buy?
⚠️ Trick: Two unknowns → need two equations (system of equations)
Solution: Let a = apples, r = oranges
a + r = 20 0.75a + 1.25r = 19
a = 20 − r → 0.75(20−r) + 1.25r = 19
15 − 0.75r + 1.25r = 19 → 0.5r = 4 → r = 8
A — 05
A number is 5 more than twice another number. Their sum is 41. What is the smaller number?
⚠️ Trick: "5 more than twice" means 2x + 5, not 2(x + 5)
Solution: Let x = smaller, y = larger
y = 2x + 5 x + y = 41
x + 2x + 5 = 41 → 3x = 36 → x = 12
Check: y = 29, sum = 41 ✓
A — 06
A tank can be filled by pipe A in 4 hours and emptied by pipe B in 6 hours. If both pipes are open, how long to fill the tank?
⚠️ Trick: Add rates, but B is negative (drains the tank)
Solution: Rate = fraction of tank per hour
A fills: +1/4 per hour B drains: −1/6 per hour
Net rate = 1/4 − 1/6 = 3/12 − 2/12 = 1/12
Time = 1 ÷ (1/12) = 12 hours
Key word: RATE problems → add fractions of the whole
A — 07
A shirt originally costs $80. It is first discounted 20%, then discounted an additional 10%. What is the final price?
⚠️ Trick: Total discount is NOT 30%. Apply each % separately.
Solution:
After 20% off: 80 × 0.80 = $64
After 10% off: 64 × 0.90 = $57.60
Effective discount = 1 − (0.8 × 0.9) = 1 − 0.72 = 28%, not 30%
A — 08
Emma is 3 times as old as her brother now. In 8 years, she will be twice as old. How old is Emma now?
⚠️ Trick: "In 8 years" → add 8 to BOTH ages before setting up ratio
Solution: Let brother = b, Emma = 3b
In 8 years: 3b + 8 = 2(b + 8)
3b + 8 = 2b + 16 → b = 8
Emma = 3 × 8 = 24
A — 09
A solution is 30% acid. How many liters of pure acid must be added to 20 liters of this solution to make it 50% acid?
⚠️ Trick: Pure acid = 100% acid. The total volume also increases.
Solution: Let x = liters of pure acid added
Acid before: 0.30 × 20 = 6 liters
After adding: (6 + x) / (20 + x) = 0.50
6 + x = 0.5(20 + x) = 10 + 0.5x
0.5x = 4 → x = 8 liters
A — 10
The length of a rectangle is 4 more than twice the width. If the perimeter is 56 cm, what is the area of the rectangle?
⚠️ Trick: Find dimensions first, then compute area — don't stop at just width or length
Solution: Let w = width, l = 2w + 4
Perimeter: 2(w + l) = 56 → w + l = 28
w + 2w + 4 = 28 → 3w = 24 → w = 8
l = 2(8) + 4 = 20
Area = 8 × 20 = 160 cm²
Geometry
10 Core Word Problems
Quick Memory Keys
SUPPLEMENTARY = 180°
COMPLEMENTARY = 90°
TRIANGLE SUM = 180°
EXTERIOR ANGLE = sum of remote interior
SIMILAR → ratios equal
PYTHAGOREAN: a²+b²=c²
G — 01
Two angles are supplementary. One angle is 28° more than the other. Find the measure of the larger angle.
⚠️ Trick: Supplementary = 180°, not 90°
Solution: Let smaller = x, larger = x + 28
x + (x + 28) = 180
2x + 28 = 180 → 2x = 152 → x = 76°
Larger = 76 + 28 = 104°
Key: SUPPLEMENTARY → adds to 180
G — 02
In triangle ABC, angle A is twice angle B, and angle C is 30° less than angle A. Find angle B.
⚠️ Trick: Triangle angles always sum to exactly 180° — set up all three in terms of one variable
Solution: Let B = x, A = 2x, C = 2x − 30
x + 2x + (2x − 30) = 180
5x − 30 = 180 → 5x = 210 → x = 42°
Check: A=84°, B=42°, C=54° → 84+42+54=180 ✓
G — 03
A ladder 13 ft long leans against a wall. The base of the ladder is 5 ft from the wall. How high up the wall does the ladder reach?
⚠️ Trick: The ladder is the hypotenuse — the longest side
Solution: a² + b² = c²
5² + h² = 13²
25 + h² = 169 → h² = 144
h = √144 = 12 ft
Pythagorean triple: 5, 12, 13 — worth memorizing!
G — 04
Two similar triangles have corresponding sides in ratio 3 : 5. The area of the smaller triangle is 27 cm². What is the area of the larger triangle?
⚠️ Trick: Area ratio = (side ratio)² — you must square the ratio
Solution: Side ratio = 3:5 → Area ratio = 3²:5² = 9:25
27/A = 9/25
A = 27 × 25/9 = 27 × 25 ÷ 9 = 75 cm²
Rule: AREA scales by square of linear ratio. Volume scales by cube.
G — 05
An exterior angle of a triangle is 115°. One of the non-adjacent interior angles is 58°. What is the other non-adjacent interior angle?
⚠️ Trick: Exterior angle = sum of the TWO non-adjacent (remote) interior angles
Solution: Exterior Angle Theorem:
Exterior = remote angle 1 + remote angle 2
115° = 58° + x → x = 57°
Memory: EXTERIOR = sum of two REMOTE interior
G — 06
A circle has circumference of 24π cm. What is the area of the circle?
⚠️ Trick: Find radius from circumference FIRST, then plug into area formula
Solution:
C = 2πr → 24π = 2πr → r = 12 cm
A = πr² = π × 12² = 144π cm²
Two-step: C → r → A. Never skip the middle step.
G — 07
A rectangular garden is 8 m × 6 m. A diagonal path cuts across it. How much shorter is the diagonal path compared to walking along two sides?
⚠️ Trick: Compare diagonal vs perimeter of 2 sides — subtract, don't just find diagonal
Solution:
Diagonal: √(8² + 6²) = √(64+36) = √100 = 10 m
Two sides: 8 + 6 = 14 m
Difference: 14 − 10 = 4 m shorter
Triple: 6, 8, 10 (= 3,4,5 scaled ×2)
G — 08
A cone has a radius of 6 cm and a height of 8 cm. What is the volume? (Use π ≈ 3.14)
⚠️ Trick: Cone volume = (1/3) × cylinder volume — students often forget the 1/3
Solution: V = (1/3)πr²h
V = (1/3) × 3.14 × 6² × 8
V = (1/3) × 3.14 × 36 × 8
V = (1/3) × 904.32 = 301.44 cm³
Cone = 1/3 Cylinder. Pyramid = 1/3 Prism. Always 1/3.
G — 09
A tree casts a shadow 15 m long. At the same time, a 2 m tall pole casts a 3 m shadow. How tall is the tree?
⚠️ Trick: This is similar triangles — set up a proportion, not addition
Solution: Similar triangles → proportional sides
height/shadow = height/shadow
2/3 = h/15 → h = (2 × 15)/3 = 10 m
Key: SIMILAR → write ratio as fraction, cross-multiply
G — 10
The surface area of a cube is 150 cm². What is the volume of the cube?
⚠️ Trick: A cube has 6 faces — divide by 6 to find one face, then take √ for side length
Solution:
Surface area = 6s² → 150 = 6s²
s² = 25 → s = 5 cm
Volume = s³ = 5³ = 125 cm³
3-step: SA ÷ 6 → find face area → √ → side → cube it