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Pre-Algebra
Order of operations · Integers · Fractions · Equations · Ratios
1
Simplify using the order of operations:
\( 3 + 4 \times 2 - (6 \div 3)^2 \)
💡 Example: \(2 + 3 \times 4 = 2 + 12 = 14\) (NOT \(5 \times 4 = 20\))
\( 3 + 4 \times 2 - (6 \div 3)^2 \)
💡 Example: \(2 + 3 \times 4 = 2 + 12 = 14\) (NOT \(5 \times 4 = 20\))
PEMDAS — your order of operations cheat code
Parentheses → Exponents → Multiply/Divide (left→right) → Add/Subtract (left→right)
2
A submarine is at \(-120\) feet. It rises \(45\) feet, then descends \(80\) feet.
What is its final depth?
💡 Rise = add a positive. Descend = subtract (go more negative).
What is its final depth?
💡 Rise = add a positive. Descend = subtract (go more negative).
Integer Number Line Rule
Up/Rise = + | Down/Drop = − | Always start from where you ARE, not zero.
3
Solve for \(x\):
\( 3x - 7 = 2x + 5 \)
💡 Goal: get all \(x\)'s on one side, all numbers on the other.
\( 3x - 7 = 2x + 5 \)
💡 Goal: get all \(x\)'s on one side, all numbers on the other.
Equation Balance Rule
Whatever you do to one side, do the SAME to the other side. Like a seesaw — keep it balanced!
4
A recipe needs \(\dfrac{2}{3}\) cup of sugar. You want to make \(2\dfrac{1}{2}\) times the recipe.
How much sugar do you need?
💡 Convert mixed number first: \(2\frac{1}{2} = \frac{5}{2}\)
How much sugar do you need?
💡 Convert mixed number first: \(2\frac{1}{2} = \frac{5}{2}\)
Mixed Number → Improper Fraction
Whole × Denominator + Numerator → put over Denominator. \(2\frac{1}{2} = \frac{2\times2+1}{2} = \frac{5}{2}\)
5
A store sells notebooks for $4 each and pens for $1.50 each. Maria spends exactly $15.
If she buys 3 notebooks, how many pens did she buy?
💡 Write an equation first: \(4n + 1.5p = 15\)
If she buys 3 notebooks, how many pens did she buy?
💡 Write an equation first: \(4n + 1.5p = 15\)
Word Problem → Equation
Identify: what you know (plug in) → what you want (variable) → solve step by step.
6
A map has a scale of \(1 \text{ inch} = 25 \text{ miles}\).
Two cities are \(3.4\) inches apart on the map. What is the actual distance?
💡 Set up a proportion: \(\dfrac{1}{25} = \dfrac{3.4}{x}\)
Two cities are \(3.4\) inches apart on the map. What is the actual distance?
💡 Set up a proportion: \(\dfrac{1}{25} = \dfrac{3.4}{x}\)
Proportion Cross-Multiply
\(\frac{a}{b} = \frac{c}{d}\) → \(a \times d = b \times c\) (cross the equals sign like an X!)
7
Solve the inequality and identify the correct graph description:
\( -3x + 6 > 15 \)
⚠ Warning: Dividing by a negative flips the inequality sign!
\( -3x + 6 > 15 \)
⚠ Warning: Dividing by a negative flips the inequality sign!
FLIP the sign when × or ÷ by NEGATIVE
\(> \) becomes \(< \) | \(\geq\) becomes \(\leq\) | Think: the arrow chases the bigger number.
8
A jacket originally costs $80. It is on sale for 35% off.
What is the sale price?
💡 Two methods: (A) Find 35% → subtract. (B) Multiply by \(1 - 0.35 = 0.65\).
What is the sale price?
💡 Two methods: (A) Find 35% → subtract. (B) Multiply by \(1 - 0.35 = 0.65\).
Percent Discount Shortcut
Sale price = Original × (1 − discount%). 35% off → multiply by 0.65. Fast!
9
Evaluate: \( (-2)^3 + (-3)^2 \)
⚠ Is \((-3)^2\) positive or negative? Think carefully about the parentheses!
⚠ Is \((-3)^2\) positive or negative? Think carefully about the parentheses!
Negative Exponent Rules
(−)^even = positive | (−)^odd = negative | \((-3)^2 = 9\) ✓ | \(-3^2 = -9\) ← no parentheses!
10
Two trains leave the same station at the same time. Train A travels east at 60 mph. Train B travels west at 75 mph.
How far apart are they after \(2\dfrac{1}{2}\) hours?
💡 Opposite directions → ADD the distances together.
How far apart are they after \(2\dfrac{1}{2}\) hours?
💡 Opposite directions → ADD the distances together.
Distance = Rate × Time (D = RT)
Same direction → SUBTRACT distances. Opposite directions → ADD distances.
G
Geometry
Angles · Triangles · Area & Perimeter · Circles · Pythagorean Theorem
11
Two lines intersect. One angle formed is \(52°\).
What are the measures of the other three angles?
💡 Vertical angles are across from each other. Supplementary angles add to 180°.
What are the measures of the other three angles?
💡 Vertical angles are across from each other. Supplementary angles add to 180°.
Intersecting Lines → 2 pairs
Vertical (across) = EQUAL | Supplementary (next to each other) = 180°
12
A right triangle has legs of length \(9\) and \(12\).
What is the length of the hypotenuse?
💡 Use: \(a^2 + b^2 = c^2\). Know the 3-4-5 family: 3×3=9, 4×3=12 → 5×3=?
What is the length of the hypotenuse?
💡 Use: \(a^2 + b^2 = c^2\). Know the 3-4-5 family: 3×3=9, 4×3=12 → 5×3=?
Pythagorean Triples — Memorize These!
3-4-5 | 5-12-13 | 8-15-17 | Multiples work too: 6-8-10, 9-12-15, 10-24-26
13
A circular pizza has a diameter of \(14\) inches.
What is the area of the pizza? (Use \(\pi \approx 3.14\))
⚠ Common mistake: students use diameter instead of radius in \(A = \pi r^2\)!
What is the area of the pizza? (Use \(\pi \approx 3.14\))
⚠ Common mistake: students use diameter instead of radius in \(A = \pi r^2\)!
Circle Formulas
Area = \(\pi r^2\) (radius squared) | Circumference = \(2\pi r = \pi d\) | r = d ÷ 2 always!
14
In triangle ABC, angle A = \(48°\) and angle B = \(73°\).
What is the measure of angle C?
Also — is this triangle acute, right, or obtuse?
💡 All three angles of any triangle ALWAYS add to 180°.
What is the measure of angle C?
Also — is this triangle acute, right, or obtuse?
💡 All three angles of any triangle ALWAYS add to 180°.
Triangle Angle Sum = 180° (Always!)
Acute: all angles < 90° | Right: one angle = 90° | Obtuse: one angle > 90°
15
A trapezoid has parallel sides of length \(6\) cm and \(10\) cm, and a height of \(4\) cm.
What is its area?
💡 Trapezoid area formula: \(A = \dfrac{1}{2}(b_1 + b_2) \times h\)
What is its area?
💡 Trapezoid area formula: \(A = \dfrac{1}{2}(b_1 + b_2) \times h\)
Trapezoid = "Average the two bases, multiply by height"
\(A = \frac{b_1 + b_2}{2} \times h\) → Think: average base × height, like a parallelogram!
16
Two parallel lines are cut by a transversal. One angle is \(110°\).
Which of the following is NOT a correct statement?
⚠ Know your angle pairs: alternate interior, co-interior (same-side), corresponding.
Which of the following is NOT a correct statement?
⚠ Know your angle pairs: alternate interior, co-interior (same-side), corresponding.
Parallel Lines Cut by Transversal
Corresponding = equal (F-shape) | Alternate interior = equal (Z-shape) | Co-interior = 180° (C-shape)
17
A rectangle has a perimeter of \(46\) cm. Its length is \(3\) more than twice its width.
Find the dimensions.
💡 Let width = \(w\). Then length = \(2w + 3\). Perimeter = \(2(\ell + w) = 46\).
Find the dimensions.
💡 Let width = \(w\). Then length = \(2w + 3\). Perimeter = \(2(\ell + w) = 46\).
Perimeter of Rectangle = 2(length + width)
Always define your variable FIRST, write the equation, THEN solve. Don't guess!
18
A point \(P\) is located at \((2, -3)\) on a coordinate plane.
It is reflected across the \(y\)-axis, then translated \(4\) units up.
What are the final coordinates?
⚠ Reflection across y-axis flips the x-coordinate's sign. Translation up adds to y.
It is reflected across the \(y\)-axis, then translated \(4\) units up.
What are the final coordinates?
⚠ Reflection across y-axis flips the x-coordinate's sign. Translation up adds to y.
Transformations Cheat Sheet
Reflect over y-axis: (x,y)→(−x,y) | Reflect over x-axis: (x,y)→(x,−y) | Translate up +k: y → y+k
19
A composite figure is made of a rectangle (\(10 \times 6\)) with a semicircle on top, centered on the 10-unit side.
What is the total area? (Use \(\pi \approx 3.14\))
💡 Semicircle radius = half of the 10-unit side = 5 units. Area = \(\frac{1}{2}\pi r^2\).
What is the total area? (Use \(\pi \approx 3.14\))
💡 Semicircle radius = half of the 10-unit side = 5 units. Area = \(\frac{1}{2}\pi r^2\).
Composite Figure Strategy
BREAK IT APART → find area of each piece → ADD them up. Never try to do it all at once!
20
Two similar triangles have sides in ratio \(3:5\).
The smaller triangle has an area of \(27\) cm².
What is the area of the larger triangle?
⚠ Tricky! Side ratio \(3:5\) does NOT mean area ratio \(3:5\). Areas scale by the ratio SQUARED.
The smaller triangle has an area of \(27\) cm².
What is the area of the larger triangle?
⚠ Tricky! Side ratio \(3:5\) does NOT mean area ratio \(3:5\). Areas scale by the ratio SQUARED.
Similar Figures Scale Rules
Side ratio = \(k\) | Perimeter ratio = \(k\) | Area ratio = \(k^2\) | Volume ratio = \(k^3\)