Mathematics Knowledge test

B. Inside the parentheses first: 1 − 4 = −3. Then the exponent: (−3)² = 9, a positive number. Next, multiply and divide: 2 × 9 = 18 and 6 ÷ 3 = 2. Last, add and subtract from left to right: 20 − 18 + 2 = 4. 40 comes from reading (−3)² as −9, 56 from working out the brackets and the exponent and then going strictly left to right (20 − 2 = 18, × 9 = 162, + 6 = 168, ÷ 3 = 56), and 0 from subtracting the 2 as well. The value is 4.

Lesson: Order of operations

C. Distribute first: 3(x − 4) means 3 times x and 3 times 4, so the left side is 3x − 12. Now 3x − 12 = x + 18. Take x away from both sides: 2x − 12 = 18. Add 12 to both sides: 2x = 30. Divide by 2: x = 15. Check: 3(15 − 4) = 33 and 15 + 18 = 33. 11 comes from multiplying only the x (3x − 4), 3 from subtracting 12 instead of adding it, and 30 from stopping before dividing by 2. x = 15.

Lesson: Solving equations

D. Get the x terms on one side. Subtract 2x from both sides: 7 − 5x ≥ −18. Subtract 7 from both sides: −5x ≥ −25. Now divide both sides by −5. Dividing by a negative number flips the sign: x ≤ 5. Check with x = 0: 7 ≥ −18 is true, and 0 is less than 5. x ≥ 5 forgets to flip the sign, and the two answers with −5 lose the minus signs when dividing −25 by −5 (a negative divided by a negative is positive). The answer is x ≤ 5.

Lesson: Inequalities

A. Line the two orders up. The second order has the same 2 sandwiches but 2 more bottles of water, and it costs $20 − $16 = $4 more. So 2 bottles cost $4, and one bottle costs $2. In the first order, 3 bottles cost 3 × $2 = $6, so the 2 sandwiches cost $16 − $6 = $10, and one sandwich costs $10 ÷ 2 = $5. $6 is the price of the 3 bottles, $8 splits the $16 as if it were all sandwiches, and $10 is the price of two sandwiches. One sandwich costs $5.

Lesson: Systems of two equations

A. Do the power first. (3x²)³ means 3x² times 3x² times 3x²: the 3 is cubed, 3 × 3 × 3 = 27, and the exponents add, 2 + 2 + 2 = 6. So (3x²)³ = 27x⁶. Now divide: 27 ÷ 9 = 3, and x⁶ ÷ x⁴ = x² (subtract the exponents). 3x⁴ comes from cubing the exponent (2³ = 8) instead of multiplying it by 3, x²/3 forgets to cube the 3, and 18x² subtracts 9 from 27 instead of dividing. The answer is 3x².

Lesson: Exponents and roots

D. FOIL multiplies every term of one bracket by every term of the other. First: 3x times 2x is 6x². Outer: 3x times 5 is 15x. Inner: −2 times 2x is −4x. Last: −2 times 5 is −10. Combine the middle terms: 15x − 4x = 11x. 6x² − 10 skips the outer and inner terms, 6x² + 19x − 10 treats −4x as +4x, and 6x² − 11x − 10 mixes up which middle term is negative. The product is 6x² + 11x − 10.

Lesson: Polynomials and FOIL

D. Call the width w. The length is w + 3, and the area is w(w + 3) = 108, so w² + 3w − 108 = 0. Look for two numbers that multiply to −108 and add to 3: 12 and −9. So (w + 12)(w − 9) = 0, and the width is 9 feet (a side can’t be −12). The length is 9 + 3 = 12 feet. Check: 9 × 12 = 108. The perimeter goes all the way around: 2 × (9 + 12) = 42 feet. 9 is the width, 12 is the length, and 21 adds only one width and one length. The perimeter is 42.

Lesson: Quadratics: factoring and solving

C. Work out each value separately, with parentheses around the negative number. f(−4) = (−4)² − 3(−4) = 16 + 12 = 28. f(2) = 2² − 3(2) = 4 − 6 = −2. Add them: 28 + (−2) = 26. −6 comes from reading (−4)² as −16, 2 from reading −3(−4) as −12, and 30 from dropping the minus sign of f(2). The answer is 26.

Lesson: Formulas and f(x)

A. Multiply the front numbers and the powers of 10 separately. 8 × 5 = 40. For the powers of 10, add the exponents: −3 + 7 = 4, so 10⁻³ times 10⁷ is 10⁴. That gives 40 × 10⁴. Scientific notation needs a front number from 1 up to (but not including) 10, so write 40 as 4 × 10¹, which adds one more power of 10: 4 × 10⁵. 4 × 10⁴ forgets that last step, 4 × 10¹¹ subtracts the exponents instead of adding them (7 − (−3) = 10, and 40 × 10¹⁰ = 4 × 10¹¹), and 13 × 10⁴ adds 8 and 5 instead of multiplying. The answer is 4 × 10⁵.

Lesson: Absolute value and scientific notation

C. One full turn rolls the cart one circumference: π × diameter = 22/7 × 28 = 88 inches. The distance is in feet, so change it to inches first: 220 × 12 = 2,640 inches. Turns: 2,640 ÷ 88 = 30. 2.5 divides 220 by 88 without changing feet to inches, 15 uses 28 as the radius (a circumference of 176 inches), and 60 multiplies π by the radius only (44 inches). The wheel makes 30 turns.

Lesson: Circles

B. The diagonal is the hypotenuse of a right triangle with legs of 80 and 60. 80² + 60² = 6,400 + 3,600 = 10,000, and √10,000 = 100 yards. (Shortcut: 60, 80, 100 is 3, 4, 5 times 20.) Along two sides she would walk 80 + 60 = 140 yards. The difference is 140 − 100 = 40 yards. 100 is the diagonal itself, 140 is the long way around, and 20 just subtracts the two sides. Her walk is 40 yards shorter.

Lesson: The Pythagorean theorem

A. Call the first angle x. The others are 2x and x + 20. The three angles of a triangle add up to 180°: x + 2x + x + 20 = 180, so 4x + 20 = 180, 4x = 160 and x = 40. The angles are 40°, 80° and 60°, and 40 + 80 + 60 = 180. 45° forgets the extra 20°, 60° is the third angle, and 80° is the largest angle. The smallest angle is 40°.

Lesson: Angles

D. Slope = change in y ÷ change in x = (−1 − 7) ÷ (3 − (−1)) = −8 ÷ 4 = −2. Put the slope and the point (−1, 7) into y = mx + b: 7 = −2 × (−1) + b, so 7 = 2 + b and b = 5. Check the other point: −2 × 3 + 5 = −1, as it should. y = 2x + 9 loses the minus sign of the slope, y = −2x + 9 reads −2 × (−1) as −2, and y = −0.5x + 6.5 divides the change in x by the change in y. The line is y = −2x + 5.

Lesson: Coordinate geometry

C. Volume of a cylinder = π × radius² × height. The radius is half the diameter: 10 ÷ 2 = 5. Radius squared: 5 × 5 = 25. Then 3.14 × 25 = 78.5 square inches for the bottom, and 78.5 × 4 = 314. 62.8 forgets to square the radius, 125.6 is the area of the side label (2 times π times 5 times 4), and 1,256 uses the diameter as the radius. The volume is 314.

Lesson: Volume and surface area

B. “At least one” is easiest to find the other way round. The chance of no 6 on one roll is 5/6, so the chance of no 6 on both rolls is 5/6 × 5/6 = 25/36. Every other result has at least one 6: 1 − 25/36 = 11/36. 1/3 adds 1/6 + 1/6, which counts a double 6 twice; 25/36 is the chance of no 6 at all; and 1/36 is the chance of two 6s. The probability is 11/36.

Lesson: Probability and counting

B. Change gallons to fluid ounces. 1 gallon = 4 quarts and 1 quart = 32 fluid ounces, so 1 gallon = 4 × 32 = 128 fluid ounces. 3 gallons = 3 × 128 = 384 fluid ounces. Cups: 384 ÷ 12 = 32. 16 uses 16 ounces for a quart (16 ounces is a pint), 48 counts 8-ounce cups instead of 12-ounce cups, and 384 is the number of ounces, not cups. It fills 32 cups.

Lesson: Units and measurement

B. Find the cost first. 2.5 pounds is 2 pounds and a half pound: 2 × $3.80 = $7.60, and half of $3.80 is $1.90. Together, $7.60 + $1.90 = $9.50. Change: $20 − $9.50 = $10.50. $9.50 is the cost, not the change; $11.90 adds 50 cents for the half pound instead of half the price; and $16.20 subtracts the price of only one pound. His change is $10.50.

Lesson: Decimals

C. Percent change = the change ÷ the price it starts from. In September the price starts at $50 and rises $80 − $50 = $30. $30 ÷ $50 = 3/5, and 3/5 = 6/10 = 60%. 37.5% divides by $80 (that is the percent of the summer drop, which starts from $80), 30% treats the $30 as a percent, and 62.5% is $50 as a part of $80. The price goes up 60%.

Lesson: Percent problems

A. Average speed = total distance ÷ total time, not the average of the two speeds. Going: 24 ÷ 12 = 2 hours. Coming back: 24 ÷ 8 = 3 hours. In all, 48 miles in 5 hours, and 48 ÷ 5 = 9.6. 10 averages the two speeds, 20 adds them, and 48 is the total distance. The average speed is 9.6.

Lesson: Distance, rate and time

C. Turn averages into totals. 5 numbers that average 18 add up to 5 × 18 = 90. 4 numbers that average 16 add up to 4 × 16 = 64. The removed number is the difference: 90 − 64 = 26. 2 subtracts the two averages, 17 averages the averages, and 34 adds them. The number removed is 26.

Lesson: Averages, remainders and units