Grade 8 Math: 10 Free Practice Questions

Answer: 15
The length is 15 centimeters. Let x be the width. The length is 2x + 5, so the perimeter equation is 2(x + 2x + 5) = 40. This simplifies to 6x + 10 = 40, so x = 5. The question asks for the length, not the width, so substitute back: 2(5) + 5 = 15. If you chose 5, you gave the width and stopped. If you chose 10, you doubled the width but forgot to add 5. Check: 2(5 + 15) = 40.

Answer: 5 x 10^7
5 x 10^7 is the largest. With scientific notation, compare the powers of ten first, because they set the size: 10^7 beats 10^6 and 10^5 no matter what the front number is. Two options share 10^7, so compare their front numbers, and 5 beats 2. Choosing 9 x 10^6 is the usual trap - the 9 looks big, but it is multiplied by a smaller power of ten. Power of ten first, front number only as a tie-break.

Answer: 288π cm³
The volume is 288π cm³. Add the two parts. The cylinder is π × 6² × 4 = 144π cm³. The hemisphere is half a sphere: (1/2) × (4/3) × π × 6³ = 144π cm³. The total is 288π cm³. If you chose 432π cm³, you used a whole sphere instead of half. If you chose 144π cm³, you left out the hemisphere. Check that the height h belongs to the cylinder only.

Answer: It is irrational because 35 is not a perfect square.
The answer is that √35 is irrational because 35 is not a perfect square: 5² = 25 and 6² = 36, so no whole number squares to 35. If you chose rational because 5 × 7 = 35, you found factors of 35, but a root is rational only when one number multiplied by itself gives 35. Being odd does not decide it either. Test against the perfect squares, not factors or parity.

Answer: The radius is 6, so V = 180π cm³
The student used the diameter as the radius. The formula needs the radius, which is half of 12 cm, so r = 6. Then V = π × 6² × 5 = 180π cm³. The student's answer is four times too large because 12² is four times 6². The 1/3 belongs only to a cone, and the height is never squared. Reading whether a figure gives the radius or the diameter is the first step of every volume problem.

Answer: 3.2 x 10^5
The answer is 3.2 x 10^5. Scientific notation needs a number from 1 up to (but not including) 10, multiplied by a power of ten. Moving the decimal point to make 3.2 takes 5 places, so the exponent is 5. If you chose 3.2 x 10^6 you miscounted by one place, which is the most common error here. Count the places twice, and remember a number smaller than 1 always has a negative exponent.

Answer: 6.71
The answer is 6.71. Squaring shows 6.70² = 44.89 and 6.71² = 45.0241, so √45 is between 6.70 and 6.71, and √45 ≈ 6.7082. The digit after the hundredths place is 8, so round up to 6.71. If you chose 6.70, you cut the digits off instead of rounding. Always look at the next digit before you drop digits.

Answer: 0
(−5)^2 = (−5)(−5) = 25, and 5^2 = 25, so 25 − 25 = 0. Each part squares a different thing: (−5)^2 squares the negative too, so it is positive 25, and 5^2 is also 25. If you chose −50 you treated (−5)^2 as negative, and 50 comes from adding the two.

Answer: The triangles are similar by AA
The triangles are similar. Triangle ABC's third angle is 180 - 40 - 65 = 75°, and triangle DEF's third angle is 180 - 65 - 75 = 40°, so both triangles have angles 40°, 65° and 75°. Equal angles are enough for similarity (AA). If you chose 'only one angle matches' you compared the angles that were GIVEN rather than finding the third one. Equal angles never prove congruence, because the sizes could still differ.

Answer: 2
The solution is x = 2. Move the x-terms to one side by subtracting 2x from both sides, leaving 7x + 3 = 17. Subtract 3 to get 7x = 14, then divide by 7 to get x = 2. If you chose 14/11, you added the x-terms instead of subtracting them. If you chose 20/7, you added 3 when you should have subtracted it. Check: both sides equal 21.