Algebra 1 & Geometry — Practice Worksheet

Algebra 1

Word problems covering equations, inequalities, functions, and systems. Read carefully — the trick is in the words!

Linear Equation

Sarah has $120. She earns $15 per hour babysitting. She wants to save at least $300. Which inequality shows the minimum number of hours $h$ she must work?

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Memory Key"At least" → greater than or equal to ≥ | "At most" → less than or equal to ≤

Example If you start with $50 and earn $10/hr and want at least $150:
$50 + 10h \ge 150$ → $10h \ge 100$ → $h \ge 10$

Explanation She starts with $120 and adds $15 per hour → total = $120 + 15h$. She wants this to be at least $300, so the inequality is $120 + 15h \ge 300$. Solving: $15h \ge 180$, so $h \ge 12$. She must work at least 12 hours.

Systems of Equations

Two friends, Alex and Jordan, together have 54 stickers. Alex has 6 more than twice Jordan's amount. How many stickers does Jordan have?

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Memory KeySET UP TWO EQUATIONS first. Let one variable = the unknown. "More than" = add. "Twice" = multiply by 2.

Example Together = 10, one has 2 more: $a+b=10$, $a=b+2$ → sub → $b+2+b=10$ → $b=4$

Explanation Let Jordan = $j$. Alex = $2j + 6$. Together: $j + (2j+6) = 54$ → $3j + 6 = 54$ → $3j = 48$ → $j = 16$. Jordan has 16 stickers. (Alex has 38.)

Rate · Distance

A train travels 240 miles in 4 hours. At the same speed, how long will it take to travel 390 miles? (Give your answer in hours.)

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Memory Key D = R × T → Rate = Distance ÷ Time → Time = Distance ÷ Rate

Example 60 miles in 2 hours → rate = 30 mph. Time for 90 miles = 90 ÷ 30 = 3 hours.

Explanation Rate = 240 ÷ 4 = 60 mph. Time = 390 ÷ 60 = 6.5 hours.

Percent · Change

A jacket originally costs $80. It is marked down 25%, then the sale price is taxed an additional 10%. What is the final price?

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Memory Key SEQUENTIAL — apply discount first, then add tax to the NEW price. Never add/subtract percents directly!

Example $100 item, 20% off → $80. Then 5% tax → $80 × 1.05 = $84.

Explanation After 25% discount: $80 \times 0.75 = \$60$. After 10% tax: $60 \times 1.10 = \$66.00$. Be careful — the 10% tax is applied to $60, not $80!

Slope · Intercept

A plumber charges a $45 flat fee plus $30 per hour. If the total bill was $165, for how many hours did the plumber work?

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Memory Key Flat fee = y-intercept (b). Per hour = slope (m). Use: $y = mx + b$

Example \$20 flat + \$10/hr, total \$60: $20 + 10x = 60$ → $x = 4$ hours

Explanation $45 + 30h = 165$ → $30h = 120$ → $h = 4$. The plumber worked for 4 hours.

Proportions

A recipe calls for 3 cups of flour to make 24 cookies. How many cups of flour are needed to make 40 cookies?

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Memory Key Cross-multiply: $\dfrac{a}{b} = \dfrac{c}{d}$ → $ad = bc$

Explanation $\dfrac{3}{24} = \dfrac{x}{40}$ → $24x = 120$ → $x = 5$. You need 5 cups of flour.

Quadratics

A ball is thrown upward. Its height in feet is given by $h(t) = -16t^2 + 32t + 6$, where $t$ is seconds. What is the maximum height of the ball?

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Memory Key Max height at vertex: $t = \dfrac{-b}{2a}$ then plug back in. For $-16t^2+32t+6$: $a=-16, b=32$

Explanation Vertex time: $t = \dfrac{-32}{2(-16)} = 1$ second. Max height: $h(1) = -16(1)^2 + 32(1) + 6 = -16+32+6 = \mathbf{22}$ feet.
Note: Answer B was intentionally tricky — always compute, don't guess!

Absolute Value

The temperature in a city must stay within 5°F of 68°F to keep food safe. Write and solve: what is the range of safe temperatures?

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Memory Key "Within __ of __" → $|x - \text{center}| \le \text{distance}$. Split into TWO inequalities!

Explanation $|T - 68| \le 5$ → $-5 \le T - 68 \le 5$ → $63 \le T \le 73$. Safe range: 63°F to 73°F.

Functions

A function $f(x) = 3x^2 - 2$. What is $f(-3)$?

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Memory Key Substitute carefully! $(-3)^2 = 9$, NOT $-9$. Negative squared = POSITIVE.

Explanation $f(-3) = 3(-3)^2 - 2 = 3(9) - 2 = 27 - 2 = \mathbf{25}$. The most common mistake is writing $(-3)^2 = -9$. Always square first, then multiply!

Mixture Problem

A chemist mixes 20 liters of a 30% acid solution with $x$ liters of a 60% acid solution to get a 50% acid solution. What is $x$?

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Memory Key MIXTURE: (liters × %) + (liters × %) = (total liters × %) → think "how much pure acid in each part"

Example 10L of 20% + x L of 80% = (10+x) L of 50%:
$10(0.20) + x(0.80) = (10+x)(0.50)$

Explanation $20(0.30) + x(0.60) = (20+x)(0.50)$
$6 + 0.6x = 10 + 0.5x$
$0.1x = 4$ → $x = \mathbf{40}$ liters.

Geometry

Angles, triangles, circles, area, volume — the most commonly missed topics. Think visually, then calculate.

Pythagorean Theorem

A 13-foot ladder leans against a wall. The bottom of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?

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Memory Key $a^2 + b^2 = c^2$. The ladder is always the hypotenuse (longest side = c). Common triplets: 3-4-5, 5-12-13, 8-15-17.

Explanation $5^2 + h^2 = 13^2$ → $25 + h^2 = 169$ → $h^2 = 144$ → $h = 12$. This is the classic 5-12-13 Pythagorean triple!

Triangle Angles

Two angles of a triangle measure 47° and 68°. What is the measure of the third angle?

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Memory Key Triangle Angle Sum = 180°. Always. Every triangle. No exceptions.

Explanation $47 + 68 + x = 180$ → $115 + x = 180$ → $x = \mathbf{65°}$.

Circle — Area & Circumference

A circular pizza has a diameter of 14 inches. What is the area of the pizza? (Use $\pi \approx 3.14$.)

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Memory Key Area = $\pi r^2$. Circumference = $2\pi r$. Diameter ÷ 2 = radius! Don't use diameter in the formula!

Explanation Radius = 14 ÷ 2 = 7 in. Area = $\pi(7)^2 = 3.14 \times 49 = \mathbf{153.86}$ in². Choice B used diameter (14) instead of radius — very common mistake!

Parallel Lines & Transversal

Two parallel lines are cut by a transversal. One angle measures 112°. What is the measure of its co-interior (same-side interior) angle?

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Memory Key Co-interior = SUPPLEMENTARY (add to 180°). Alternate interior = EQUAL. Corresponding = EQUAL.

Explanation Co-interior angles are supplementary: $112 + x = 180$ → $x = \mathbf{68°}$. (They are on the same side of the transversal, between the parallel lines.)

Volume — Cylinder

A cylindrical water tank has a radius of 4 m and a height of 10 m. What is its volume? (Use $\pi \approx 3.14$.)

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Memory Key V = $\pi r^2 h$. Think: area of circle × height = volume. Radius, NOT diameter!

Explanation $V = \pi r^2 h = 3.14 \times 4^2 \times 10 = 3.14 \times 16 \times 10 = \mathbf{502.4}$ m³. Choice D used diameter (8) instead of radius.

Similar Triangles

Two similar triangles have sides in ratio 3:5. If the smaller triangle has a perimeter of 36 cm, what is the perimeter of the larger triangle?

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Memory Key Similar = same SHAPE, different SIZE. Perimeters scale by the SAME ratio as the sides.

Explanation $\dfrac{3}{5} = \dfrac{36}{P}$ → $P = \dfrac{36 \times 5}{3} = \mathbf{60}$ cm.

Area — Composite Shape

A shape is formed by a rectangle (6 m × 4 m) with a semicircle on top (diameter = 6 m). What is the total area? (Use $\pi \approx 3.14$.)

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Memory Key Composite = ADD parts. Semicircle area = $\dfrac{\pi r^2}{2}$. Diameter 6 → radius 3.

Explanation Rectangle: $6 \times 4 = 24$ m². Semicircle (r=3): $\dfrac{3.14 \times 9}{2} = 14.13$ m². Total: $24 + 14.13 = \mathbf{38.13}$ m². Answer A is correct!

Exterior Angles

An exterior angle of a triangle measures 130°. The two non-adjacent interior angles are $x$ and $x + 20$. What is $x$?

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Memory Key Exterior Angle = sum of the two NON-ADJACENT interior angles. (This is the Exterior Angle Theorem.)

Explanation $x + (x+20) = 130$ → $2x + 20 = 130$ → $2x = 110$ → $x = \mathbf{55°}$. The other interior angle is 75°. Check: 55 + 75 = 130 ✓

Surface Area — Rectangular Prism

A box has length 5 cm, width 3 cm, and height 4 cm. How much wrapping paper is needed to cover the entire box? (Find the surface area.)

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Memory Key SA = $2(lw + lh + wh)$. There are 3 pairs of faces — count each twice!

Explanation $SA = 2(5\times3 + 5\times4 + 3\times4) = 2(15+20+12) = 2(47) = \mathbf{94}$ cm².

G10

Coordinate Geometry

Point $A = (1, 2)$ and point $B = (7, 10)$. What is the length of segment $AB$?

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Memory Key Distance = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. It's just the Pythagorean theorem on a grid!

Example $A=(0,0), B=(3,4)$: $\sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5$

Explanation $\sqrt{(7-1)^2 + (10-2)^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = \mathbf{10}$. This is the 6-8-10 Pythagorean triple (a scaled 3-4-5)!