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Grade 7 Prep Guide

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Adaptive Math Test — Grade 7 Prep Guide

A comprehensive, topic-by-topic study guide for the Grade 7 adaptive math assessment. None of the practice problems below are taken from the test bank — they are fresh examples designed to teach the same skills.

For each topic you'll find:

  • A short explanation of the key idea
  • Multiple worked examples (problems solved step by step)
  • Common mistakes to watch out for
  • Practice problems with answers at the end of each section

Table of Contents

Ratios & Proportional Relationships

  1. Proportional Relationships
  2. Unit Rates with Fractions
  3. Constant of Proportionality
  4. Scale Factors and Scale Drawings

Percent 5. Percent of a Number 6. Percent Change 7. Tax, Tip, Discount, Markup 8. Simple Interest

Number System 9. Operations with Negative Numbers 10. Operations with Fractions 11. Operations with Decimals 12. Ordering Rational Numbers

Expressions & Equations 13. Combining Like Terms & Distributive Property 14. Two-Step Equations 15. Inequalities

Geometry 16. Angle Relationships 17. Area and Circumference of Circles 18. Surface Area & Volume

Statistics & Probability 19. Sampling and Inferences 20. Probability

1. Proportional Relationships

Key Concepts

Two quantities y and x are proportional if y = kx for some constant k (the constant of proportionality). The graph of a proportional relationship is a straight line through the origin.

To test: y/x should give the same value for every pair of (x, y).

Worked Examples

Example 1. Is (3, 12), (5, 20), (8, 32) proportional? Solution. 12/3 = 4, 20/5 = 4, 32/8 = 4. Yes; k = 4.

Example 2. Find k for y = 7 when x = 14. Solution. k = y/x = 7/14 = 1/2.

Example 3. A car uses 9 gallons to drive 252 miles. Miles per gallon (k)? Solution. 252 ÷ 9 = 28 mpg.

Example 4. Write the equation for the proportional relationship where y = 18 when x = 6. Solution. k = 3 → y = 3x.

Common Mistakes

  • Treating any line as proportional — only lines through the origin are.
  • Calculating k = x/y instead of y/x.

Practice

  1. Is {(2, 6), (4, 12), (5, 14)} proportional?
  2. Find k if y = 60 when x = 5.
  3. 4 books cost $32. Write y = kx.
  4. If y = 15 when x = 5, find y when x = 12.

Answers: 1. No (last pair gives k=2.8) | 2. 12 | 3. y = 8x | 4. 36

2. Unit Rates with Fractions

Key Concepts

Unit rates can have fractions in either the numerator or denominator. Compute by dividing — and remember dividing by a fraction means multiplying by its reciprocal.

Worked Examples

Example 1. 1/2 cup of sugar in 3/4 cup of recipe. Sugar per cup of recipe? Solution. 1/2 ÷ 3/4 = 1/2 × 4/3 = 2/3 cup of sugar.

Example 2. A walker covers 3/4 mile in 1/4 hour. Speed? Solution. 3/4 ÷ 1/4 = 3/4 × 4 = 3 mph.

Example 3. $5 buys 2/3 pound. Cost per pound? Solution. 5 ÷ 2/3 = 5 × 3/2 = $7.50/lb.

Example 4. A snail moves 1/8 foot in 1/2 minute. Ft per minute? Solution. 1/8 ÷ 1/2 = 1/8 × 2 = 1/4 ft/min.

Common Mistakes

  • Forgetting to flip the divisor.
  • Mixing units (use the same unit on top vs bottom).

Practice

  1. 2/3 cup juice mixed with 1/4 cup water. Juice per cup of water?
  2. 3/8 mile in 1/2 hour — speed?
  3. A pump fills 1/3 tank in 2/3 hour. Tanks per hour?

Answers: 1. 8/3 (≈2.67) | 2. 3/4 mph | 3. 1/2

3. Constant of Proportionality

Key Concepts

In y = kx, k is the slope (and the value of y when x = 1). From a table or graph, find k by dividing y by x for any point.

Worked Examples

Example 1. A table: (1, 5), (2, 10), (3, 15). Find k. Solution. k = 5, equation y = 5x.

Example 2. Graph passes through (4, 12) and (0, 0). Find k. Solution. 12/4 = 3.

Example 3. From y = 9x, what's y when x = 4? Solution. 9(4) = 36.

Common Mistakes

  • Forgetting that the graph must pass through (0, 0).
  • Reversing x and y when reading from a table.

Practice

  1. Find k from (8, 56).
  2. y = 6x. Find y when x = 11.
  3. Find k from (1/2, 4).

Answers: 1. 7 | 2. 66 | 3. 8

4. Scale Factors and Scale Drawings

Key Concepts

A scale factor is the ratio of lengths between a scale drawing and the actual object. Areas scale by the square of the scale factor; volumes by the cube.

Worked Examples

Example 1. A drawing uses 1 inch = 5 feet. A wall is 4 inches in the drawing. Real length? Solution. 4 × 5 = 20 feet.

Example 2. Scale factor 1:50. Window is 2 cm on the plan. Real width? Solution. 2 × 50 = 100 cm = 1 m.

Example 3. Two similar rectangles have scale factor 3. Original area 8 sq in. New area? Solution. 8 × 3² = 72 sq in.

Example 4. A model car is 1/20 of the real car. Real car is 4.4 m long. Model length in cm? Solution. 4.4 / 20 = 0.22 m = 22 cm.

Common Mistakes

  • Applying linear scale to area or volume without squaring/cubing.
  • Mixing units (inches vs feet).

Practice

  1. Scale 1:25. Plan length 3 cm — actual?
  2. Similar triangles with ratio 4:1. Smaller area 5; larger area?
  3. A 1:200 map shows a road as 7 cm. Actual length?

Answers: 1. 75 cm | 2. 80 | 3. 1400 cm = 14 m

5. Percent of a Number

Key Concepts

p% of n = (p/100) × n.

Useful conversions: 1/4 = 25%, 1/5 = 20%, 1/3 ≈ 33%, 1/8 = 12.5%.

Worked Examples

Example 1. 35% of 80. Solution. 0.35 × 80 = 28.

Example 2. 6% of 250. Solution. 0.06 × 250 = 15.

Example 3. 110% of 90. Solution. 1.10 × 90 = 99.

Example 4. Find 3 1/2 % of 200. Solution. 0.035 × 200 = 7.

Common Mistakes

  • Forgetting to convert percent to decimal.
  • Misplacing the decimal (e.g., 6% = 0.06, not 0.6).

Practice

  1. 45% of 120
  2. 7% of 350
  3. 0.5% of 800
  4. 200% of 35

Answers: 1. 54 | 2. 24.5 | 3. 4 | 4. 70

6. Percent Change

Key Concepts

Percent change = (new − old) / old × 100.

  • If positive, it's a percent increase.
  • If negative, it's a percent decrease.

Worked Examples

Example 1. A price rose from $40 to $50. Percent increase? Solution. (50 − 40)/40 = 10/40 = 0.25 = 25%.

Example 2. A population dropped from 5,000 to 4,500. Percent decrease? Solution. (4500 − 5000)/5000 = −500/5000 = −10% (10% decrease).

Example 3. A jacket was $80 and is now $60. Percent off? Solution. 20/80 = 25% off.

Example 4. Stock went from $25 to $40. Percent change? Solution. 15/25 = 60% increase.

Common Mistakes

  • Dividing by the new amount instead of the old.
  • Reporting raw difference instead of a percent.

Practice

  1. From 60 to 75.
  2. From 200 to 150.
  3. A salary rose from $30k to $33k.
  4. A test score went from 80 to 92.

Answers: 1. 25% inc | 2. 25% dec | 3. 10% inc | 4. 15% inc

7. Tax, Tip, Discount, Markup

Key Concepts

  • Tax / tip / markup: multiply by (1 + rate). E.g., 8% tax → multiply price by 1.08.
  • Discount: multiply by (1 − rate). E.g., 20% off → multiply by 0.80.

Worked Examples

Example 1. $40 meal with 18% tip. Total? Solution. 40 × 1.18 = $47.20.

Example 2. $25 item with 6% sales tax. Total? Solution. 25 × 1.06 = $26.50.

Example 3. $80 jacket, 30% off. Sale price? Solution. 80 × 0.70 = $56.

Example 4. A store buys an item for $50 and marks it up 40%. Selling price? Solution. 50 × 1.40 = $70.

Common Mistakes

  • Multiplying by the rate alone instead of (1 ± rate) when finding the total.
  • Combining tax and tip incorrectly (apply both to the original or the pre-tax total — match the problem's instructions).

Practice

  1. $36 meal, 20% tip.
  2. $90 item, 7% tax.
  3. $120 dress, 15% off.
  4. Cost $25, 60% markup.

Answers: 1. $43.20 | 2. $96.30 | 3. $102 | 4. $40

8. Simple Interest

Key Concepts

I = P × r × t, where P = principal, r = annual rate (decimal), t = time in years.

Total amount: A = P + I = P(1 + rt).

Worked Examples

Example 1. $500 at 4% simple interest for 3 years. Interest? Solution. 500 × 0.04 × 3 = $60.

Example 2. $1,200 at 6% for 2 years. Total amount? Solution. 1200 × 1.12 = $1,344.

Example 3. A loan of $800 earns $96 over 4 years. Rate? Solution. 96 = 800 × r × 4 → r = 0.03 = 3%.

Common Mistakes

  • Using percent (3%) directly instead of decimal (0.03).
  • Forgetting to use years (months ÷ 12).

Practice

  1. I for $1,000 at 5% for 2 years.
  2. I for $300 at 8% for 6 months.
  3. Total amount: $500 at 3% for 5 years.

Answers: 1. $100 | 2. $12 | 3. $575

9. Operations with Negative Numbers

Key Concepts

  • Same signs → positive product/quotient.
  • Different signs → negative product/quotient.
  • Adding a negative = subtracting; subtracting a negative = adding.

Worked Examples

Example 1. −5 + (−7) = −12.

Example 2. −9 − (−4) = −9 + 4 = −5.

Example 3. (−6)(−4) = 24.

Example 4. 35 ÷ (−7) = −5.

Example 5. −2 × 3 × (−4) = 24 (two negatives → positive).

Common Mistakes

  • Counting signs in a product wrong: even number of negatives → +, odd → −.
  • Treating subtraction without converting to addition.

Practice

  1. −8 + 13
  2. (−4) × (−9)
  3. 20 − (−5)
  4. (−2)³

Answers: 1. 5 | 2. 36 | 3. 25 | 4. −8

10. Operations with Fractions

Key Concepts

  • Add / Subtract: common denominator first.
  • Multiply: top × top, bottom × bottom; simplify.
  • Divide: multiply by the reciprocal.
  • Mixed numbers: convert to improper fractions before multiplying or dividing.

Worked Examples

Example 1. 2/3 + 1/4 Solution. LCD 12: 8/12 + 3/12 = 11/12.

Example 2. 3/5 × 10/9 Solution. 30/45 = 2/3.

Example 3. 1 1/4 ÷ 5/8 Solution. 5/4 × 8/5 = 40/20 = 2.

Example 4. 2 1/3 + 1 1/2 Solution. 7/3 + 3/2 = 14/6 + 9/6 = 23/6 = 3 5/6.

Common Mistakes

  • Adding numerators AND denominators (1/2 + 1/3 ≠ 2/5).
  • Forgetting to convert mixed numbers.

Practice

  1. 5/6 − 1/4
  2. 4/5 × 15/8
  3. 7/8 ÷ 14/3
  4. 3 1/2 − 1 3/4

Answers: 1. 7/12 | 2. 3/2 | 3. 3/16 | 4. 1 3/4

11. Operations with Decimals

Key Concepts

  • Add/Sub: align decimal points.
  • Multiply: ignore decimals, multiply, then count total decimal places in factors.
  • Divide: shift the divisor to a whole number; shift the dividend the same.

Worked Examples

Example 1. 0.345 × 0.2 Solution. 345 × 2 = 690; 3 + 1 = 4 places → 0.0690 = 0.069.

Example 2. 12.6 ÷ 0.3 Solution. 126 ÷ 3 = 42.

Example 3. 5.27 + 13.4 Solution. 5.27 + 13.40 = 18.67.

Example 4. 9 − 2.475 Solution. 9.000 − 2.475 = 6.525.

Common Mistakes

  • Misaligning the decimal column.
  • Wrong count of decimal places in products.

Practice

  1. 0.8 × 0.05
  2. 7.5 ÷ 0.25
  3. 4.2 + 0.367
  4. 10 − 3.65

Answers: 1. 0.04 | 2. 30 | 3. 4.567 | 4. 6.35

12. Ordering Rational Numbers

Key Concepts

Convert each number to the same form (all decimals, or all fractions with a common denominator) and compare. On a number line, smaller values are to the left.

Worked Examples

Example 1. Order from smallest: −3/4, 0.5, −1.2, 1/3. Solution. Decimals: −0.75, 0.5, −1.2, 0.33 → −1.2, −0.75, 0.33, 0.5.

Example 2. Is −5/8 greater than −3/4? Solution. −0.625 vs −0.75; −0.625 is to the right → Yes.

Example 3. Between 7/10 and 3/4, which is bigger? Solution. 0.7 vs 0.75 → 3/4.

Common Mistakes

  • Forgetting that with negatives, the number with the larger absolute value is smaller.

Practice

  1. Order from smallest: 0.6, 2/3, 0.65.
  2. Compare: −0.4 vs −1/2.
  3. Order: −5/6, −3/4, −1/2, 0.

Answers: 1. 0.6 < 0.65 < 2/3 | 2. −0.4 > −1/2 | 3. −5/6, −3/4, −1/2, 0

13. Combining Like Terms & Distributive Property

Key Concepts

Like terms have the same variable raised to the same power. Combine them by adding/subtracting the coefficients.

Distributive property: a(b + c) = ab + ac.

Worked Examples

Example 1. Simplify 3x + 5 − x + 2. Solution. (3x − x) + (5 + 2) = 2x + 7.

Example 2. Simplify 4(2x + 3) − 5. Solution. 8x + 12 − 5 = 8x + 7.

Example 3. Simplify −2(x − 5) + 3x. Solution. −2x + 10 + 3x = x + 10.

Example 4. Combine: 5a − 3b + 2a + 4b. Solution. (5a + 2a) + (−3b + 4b) = 7a + b.

Common Mistakes

  • Distributing only to the first term: 4(2x + 3) ≠ 8x + 3.
  • Mishandling the sign when distributing a negative.

Practice

  1. Simplify 7y − 3 + 2y + 8.
  2. Distribute: 3(4x − 5).
  3. Simplify −(2x − 7) + x.
  4. 2(3a + 4) − 5(a − 1).

Answers: 1. 9y + 5 | 2. 12x − 15 | 3. −x + 7 | 4. a + 13

14. Two-Step Equations

Key Concepts

To solve, undo addition/subtraction first, then undo multiplication/division.

ax + b = c → subtract b, then divide by a.

Worked Examples

Example 1. Solve 2x + 5 = 15. Solution. 2x = 10 → x = 5.

Example 2. Solve 3x − 7 = 14. Solution. 3x = 21 → x = 7.

Example 3. Solve x/4 + 6 = 10. Solution. x/4 = 4 → x = 16.

Example 4. Solve −2x + 9 = 1. Solution. −2x = −8 → x = 4.

Example 5. Word: A taxi charges $3 plus $2 per mile. If the bill was $17, how many miles? Solution. 2m + 3 = 17 → m = 7.

Common Mistakes

  • Dividing first instead of doing addition/subtraction.
  • Sign errors when dividing by a negative.

Practice

  1. 5x − 4 = 21
  2. x/3 + 2 = 9
  3. 4x + 11 = 3
  4. 2(x − 5) = 12

Answers: 1. x = 5 | 2. x = 21 | 3. x = −2 | 4. x = 11

15. Inequalities

Key Concepts

Solve like an equation, with one important rule: multiplying or dividing both sides by a negative flips the inequality sign.

Graph on a number line: open circle for <, > and closed circle for ≤, ≥.

Worked Examples

Example 1. Solve 3x − 4 > 11. Solution. 3x > 15 → x > 5.

Example 2. Solve −2x ≤ 8. Solution. Divide by −2 (flip!): x ≥ −4.

Example 3. Solve 5 − x < 12. Solution. −x < 7 → x > −7.

Example 4. A movie ticket costs $9 and snacks cost x. Total ≤ $20. Inequality for snack budget? Solution. 9 + x ≤ 20 → x ≤ $11.

Common Mistakes

  • Forgetting the flip when dividing/multiplying by a negative.
  • Using the wrong type of circle.

Practice

  1. 2x + 3 ≥ 13
  2. −4x < 20
  3. (x − 5)/2 ≥ 4
  4. 7 − 2x > 1

Answers: 1. x ≥ 5 | 2. x > −5 | 3. x ≥ 13 | 4. x < 3

16. Angle Relationships

Key Concepts

  • Supplementary angles sum to 180°.
  • Complementary angles sum to 90°.
  • Vertical angles (formed by two intersecting lines) are equal.
  • Angles on a straight line sum to 180°.
  • The angles of a triangle sum to 180°.

Worked Examples

Example 1. Two supplementary angles. One is 70°. Other? Solution. 180 − 70 = 110°.

Example 2. Complementary to 33° is 57°.

Example 3. Two vertical angles. One is 4x and the other is 3x + 25. Find x. Solution. 4x = 3x + 25 → x = 25.

Example 4. Triangle angles: 50°, 60°, x. Find x. Solution. 180 − 110 = 70°.

Common Mistakes

  • Confusing supplementary (180°) and complementary (90°).
  • Assuming all triangles have a 90° — they don't.

Practice

  1. Complement of 22°.
  2. Supplement of 145°.
  3. Two vertical angles: 5x and 70°. Find x.
  4. Triangle angles: 35°, 85°, x.

Answers: 1. 68° | 2. 35° | 3. x = 14 | 4. 60°

17. Area and Circumference of Circles

Key Concepts

  • Circumference: C = πd = 2πr.
  • Area: A = πr².

Use π ≈ 3.14 (or 22/7) unless told to leave answers in terms of π.

Worked Examples

Example 1. Circumference of a circle with r = 5. Solution. 2π(5) = 10π ≈ 31.4.

Example 2. Area of a circle with r = 7. Solution. π(7)² = 49π ≈ 153.86.

Example 3. Diameter = 12. Find circumference. Solution. π × 12 = 12π ≈ 37.68.

Example 4. Area of a semicircle (half-circle) with r = 6. Solution. (1/2)π(6)² = 18π ≈ 56.52.

Common Mistakes

  • Using diameter where radius is needed (and vice versa).
  • Forgetting to square the radius for area.

Practice

  1. C if r = 10 (use 3.14).
  2. A if r = 4.
  3. C if d = 14 (use 22/7).
  4. A of a quarter-circle, r = 8.

Answers: 1. 62.8 | 2. 16π ≈ 50.24 | 3. 44 | 4. 16π ≈ 50.24

18. Surface Area & Volume

Key Concepts

  • Prism volume: V = (area of base) × height.
  • Cylinder volume: V = πr²h.
  • Cylinder surface area: SA = 2πr² + 2πrh.

Worked Examples

Example 1. V of a triangular prism with base area 12 and height 10. Solution. 12 × 10 = 120.

Example 2. V of a cylinder r = 3, h = 8. Solution. π(9)(8) = 72π ≈ 226.

Example 3. SA of cylinder r = 5, h = 10. Solution. 2π(25) + 2π(5)(10) = 50π + 100π = 150π ≈ 471.

Example 4. A cube has volume 216. Side length? Solution. ³√216 = 6.

Common Mistakes

  • Using diameter for r in volume/SA formulas.
  • Confusing volume (cubic) and surface area (square) units.

Practice

  1. V of cylinder r = 2, h = 5.
  2. V of cube with side 7.
  3. SA of cylinder r = 4, h = 3.
  4. V of triangular prism, base area 18, h = 6.

Answers: 1. 20π ≈ 62.8 | 2. 343 | 3. 56π ≈ 175.8 | 4. 108

19. Sampling and Inferences

Key Concepts

  • A population is everything; a sample is a subset.
  • A random sample gives each member of the population an equal chance of being chosen — that's how you avoid bias.
  • Use sample proportions to estimate population proportions.

Worked Examples

Example 1. A random sample of 50 students shows 30 like pizza. What proportion is that? Solution. 30/50 = 0.6 or 60%.

Example 2. If 60% of 50 like pizza, estimate how many of 1,000 students like pizza. Solution. 0.6 × 1000 = 600.

Example 3. Surveying only the school's basketball team about a school-wide question is... Solution. Biased — not representative.

Example 4. Two samples of 20 from the same population both show ~60% like pizza. Estimate is more reliable.

Common Mistakes

  • Generalising from a non-random or small sample.
  • Confusing the sample with the population.

Practice

  1. 20 out of 80 in a sample wear glasses. Proportion?
  2. If 25% of a 200-person sample plays soccer, estimate for population of 5,000.
  3. Sampling only adults to ask about kids' shows — bias type?

Answers: 1. 0.25 = 25% | 2. 1,250 | 3. Sampling bias (non-representative)

20. Probability

Key Concepts

  • P(event) = favorable outcomes / total outcomes (when equally likely).
  • P always between 0 and 1.
  • Complement: P(not A) = 1 − P(A).
  • Independent events: P(A and B) = P(A) × P(B).

Worked Examples

Example 1. Probability of rolling a 4 on a fair die. Solution. 1/6.

Example 2. P(even number on a die). Solution. 3/6 = 1/2.

Example 3. P(not a 5 on a die). Solution. 1 − 1/6 = 5/6.

Example 4. P(heads then heads on two flips). Solution. 1/2 × 1/2 = 1/4.

Example 5. A bag has 3 red, 5 blue, 2 green. P(red)? Solution. 3/10.

Common Mistakes

  • Forgetting that probabilities sum to 1.
  • Multiplying for "or" events (should add, with care about overlap).

Practice

  1. P(rolling a number > 4 on a die).
  2. P(drawing a heart from a 52-card deck).
  3. P(two tails in two flips).
  4. P(not rolling a 1 or 2 on a die).

Answers: 1. 2/6 = 1/3 | 2. 13/52 = 1/4 | 3. 1/4 | 4. 4/6 = 2/3

That's Grade 7. Focus on percent change, two-step equations, angle relationships, and probability — they show up most often on the test. Practice mixing positive and negative numbers until the sign rules are second nature. Good luck!