Numbers and the four operations
- 1.In the number 5,308,472, what is the 3 worth?
- 2.8,005 − 3,467 =
- 3.36 × 24 =
- 4.12 + 3 × (10 − 6) ÷ 2 =
Signed Numbers (positives & negatives)
- 5.−7 + 12 =
- 6.4 − (−9) =
- 7.−6 × (−5) =
Fractions
- 8.23 + 14 =
- 9.34 × 25 =
- 10.12 ÷ 16 =
- 11.Which is larger: 58 or 35?
Decimals
- 12.6.4 − 2.75 =
- 13.0.6 × 0.35 =
- 14.Write 38 as a decimal.
- 15.A phone plan costs $18.75 a month. What do twelve months cost?
Percents, ratios, and proportion
- 16.What is 15% of 60?
- 17.A $72 coat is marked 25% off. What is the sale price?
- 18.What is 18 out of 24 as a percent?
- 19.On a map, 2 inches stands for 15 miles. How many miles does 7 inches stand for?
Measurement and units
- 20.How many ounces are in 4 pounds?
- 21.A recipe calls for 3 quarts of stock. How many cups is that?
- 22.A shelf is 2 feet 7 inches long. How many inches is that?
Geometry
- 23.A triangle has a base of 10 inches and a height of 6 inches. What is its area?
- 24.A circle has a radius of 4 cm. What is its area? Use 3.14 for π.
- 25.A right triangle has legs of 9 and 12. How long is the hypotenuse?
- 26.Two lines cross. One of the four angles is 65°. What is the angle directly opposite it?
- 27.A box is 6 inches long, 4 inches wide, and 3 inches tall. What is its volume?
- 28.A can has a radius of 3 inches and a height of 7 inches. What is its volume? Use 3.14 for π.
Data, graphs, and statistics
- 29.Coats sold
How many more coats were sold on Tuesday than on Thursday?Day Coats Monday 34 Tuesday 51 Wednesday 47 Thursday 29 - 30.Find the mean of 12, 7, 9, 16.
- 31.Find the median of 8, 3, 11, 6, 9.
- 32.Find the range of 14, 5, 22, 9.
Probability and counting
- 33.A bag holds 4 red marbles, 3 blue, and 5 green. You draw one without looking. What is the probability it is blue?
- 34.The forecast says there is a 30% chance of rain. What is the probability it does not rain?
- 35.A diner offers 4 sandwiches and 3 soups. How many different sandwich-and-soup lunches can be made?
Exponents, roots, and scientific notation
- 36.43 =
- 37.√81 =
- 38.Write as a single power: x5 × x3
- 39.What is 50?
- 40.Write 6,200,000 in scientific notation.
Expressions and polynomials
- 41.Simplify: 5x + 3 − 2x + 7
- 42.Multiply out: 4(2x − 5)
- 43.Multiply out: (x + 3)(x − 2)
Equations and inequalities
- 44.Solve for x: 3x + 7 = 22
- 45.Solve for x: 5x − 4 = 2x + 14
- 46.Solve: 2x − 3 > 9
- 47.A plumber charges $60 to come out, plus $45 an hour. The bill came to $285. How many hours did the job take?
Lines, slope, and systems
- 48.What is the slope of the line through (2, 3) and (6, 11)?
- 49.For the line y = −3x + 5, what are the slope and the y-intercept?
- 50.The line passes through the two marked points. What is its slope?
- 51.Solve the system: x + y = 12 and x − y = 4
Functions and the coordinate plane
- 52.If f(x) = 4x − 3, what is f(5)?
- 53.If f(x) = 2x + 9, for which x is f(x) = 25?
- 54.What are the coordinates of point A?
- 55.What is the midpoint of the segment from A (−2, 1) to B (6, 7)?
Or check one family at a time, with the button under it.
Where to go next
| Family | Questions | Right | Where to go next | Practice |
|---|---|---|---|---|
| 1. Numbers and the four operations | 1–4 | of 4 | Go to Numbers and the four operationsseven quizzes, the times tables, and what to look up elsewhere | |
| 2. Signed numbers | 5–7 | of 3 | Go to Signed Numbers (positives & negatives)three quizzes, Negative Mazarine!, and Diagnostic 2 | |
| 3. Fractions | 8–11 | of 4 | Go to Fractionsfive quizzes, and Diagnostic 1 for a closer look | |
| 4. Decimals | 12–15 | of 4 | Go to Decimalsfour quizzes, and Diagnostic 1 for a closer look | |
| 5. Percents, ratios, and proportion | 16–19 | of 4 | Go to Percents, ratios, and proportionsix quizzes, in two legs | |
| 6. Measurement and units | 20–22 | of 3 | Go to Measurement and unitsQuiz 26, and what to look up elsewhere | |
| 7. Geometry | 23–28 | of 6 | Go to GeometryQuiz 27, Circles and Equations, The Third Side, and Diagnostic 3 | |
| 8. Data, graphs, and statistics | 29–32 | of 4 | Go to Data, graphs, and statisticsQuiz 28, and what to look up elsewhere | |
| 9. Probability and counting | 33–35 | of 3 | Go to Probability and countingnothing written for it yet — the page says where to go meanwhile | |
| 10. Exponents, roots, and scientific notation | 36–40 | of 5 | Go to Exponents, roots, and scientific notationfive quizzes, two apps, and a chart to print | |
| 11. Expressions and polynomials | 41–43 | of 3 | Go to Expressions and polynomialsLike With Like, and Diagnostic 3 | |
| 12. Equations and inequalities | 44–47 | of 4 | Go to Equations and inequalitiesthree rooms, a practice sheet, and Diagnostic 3 | |
| 13. Lines, slope, and systems | 48–51 | of 4 | Go to Lines, slope, and systemsLines on the Grid, and Diagnostic 3 | |
| 14. Functions and the coordinate plane | 52–55 | of 4 | Go to Functions and the coordinate planeLines on the Grid, and Diagnostic 3 |
The quiz series is at thepeoplesshare.org/ged/math.
The key prints only when it is showing, and it starts on its own page. After you check a family, each question you missed offers its own reasoning with the Why? button.
Answers, with the reasoning in brief
- 1.300,000. The 3 stands in the hundred-thousands place, so it is worth three hundred thousand. The whole number reads: five million, three hundred eight thousand, four hundred seventy-two.
- 2.4,538. You cannot take 7 from 5, and the tens and hundreds are zeros, so the borrow comes all the way from the thousands: 8,005 becomes 7 thousands, 9 hundreds, 9 tens, and 15 ones. Then 15 − 7 = 8, 9 − 6 = 3, 9 − 4 = 5, 7 − 3 = 4. Check: 4,538 + 3,467 = 8,005.
- 3.864. Break 24 into 20 and 4: 36 × 20 = 720 and 36 × 4 = 144; 720 + 144 = 864.
- 4.18. Parentheses first: 10 − 6 = 4. Then multiplying and dividing, left to right: 3 × 4 = 12, and 12 ÷ 2 = 6. Adding last: 12 + 6 = 18.
- 5.5. Start at −7 and go up 12. The first 7 steps bring you to 0; the other 5 take you to 5.
- 6.13. Subtracting a negative is the same as adding a positive: 4 − (−9) = 4 + 9 = 13.
- 7.30. Two negatives multiplied give a positive: 6 × 5 = 30, and the two minus signs cancel each other.
- 8.1112. Thirds and quarters do not match. Twelfths hold both: 23 = 812 and 14 = 312. Then 8 + 3 = 11 twelfths.
- 9.310. Multiply straight across: 3 × 2 = 6 on top, 4 × 5 = 20 underneath. 620 reduces to 310.
- 10.3. The question is how many sixths fit inside a half. A half is three sixths, so the answer is 3. The rule says the same thing: turn the second fraction over and multiply, 12 × 61 = 62 = 3.
- 11.58. In fortieths, 58 = 2540 and 35 = 2440. Twenty-five parts beat twenty-four.
- 12.3.65. Line up the decimal points and write 6.4 as 6.40 so the places match: 6.40 − 2.75 = 3.65.
- 13.0.21. Multiply as whole numbers: 6 × 35 = 210. Then count the decimal places in the question — one and two, three in all — and put three into the answer: 0.210, which is 0.21.
- 14.0.375. A fraction is a division: 3 ÷ 8. Eight does not go into 3, so 3.000 ÷ 8 = 0.375.
- 15.$225.00. 12 × 18.75: 12 × 18 = 216 and 12 × 0.75 = 9; 216 + 9 = 225.
- 16.9. 10% of 60 is 6, and 5% is half of that, 3. Together: 6 + 3 = 9.
- 17.$54. 25% is a quarter, and a quarter of 72 is 18. Take it off: 72 − 18 = 54.
- 18.75%. 1824 reduces to 34, and three quarters is 75 per hundred.
- 19.52.5 miles. One inch stands for 15 ÷ 2 = 7.5 miles, so 7 inches stands for 7 × 7.5 = 52.5 miles. Set up as a proportion it reads 2/15 = 7/x.
- 20.64. A pound is 16 ounces: 4 × 16 = 64.
- 21.12. A quart is 4 cups: 3 × 4 = 12.
- 22.31. Two feet is 2 × 12 = 24 inches. Add the 7: 24 + 7 = 31.
- 23.30 square inches. Area of a triangle is half of base × height: 10 × 6 = 60, and half of 60 is 30.
- 24.50.24 square cm. Area of a circle is π × r²: 4 × 4 = 16, and 3.14 × 16 = 50.24. The formula is on the sheet; choosing it is the work.
- 25.15. Pythagorean theorem: a² + b² = c². 9² + 12² = 81 + 144 = 225, and √225 = 15.
- 26.65°. When two lines cross, the angles opposite each other are equal. (Each pair of neighbors adds to 180°, which is what makes it so.)
- 27.72 cubic inches. Volume of a box is length × width × height: 6 × 4 = 24, and 24 × 3 = 72.
- 28.197.82 cubic inches. Volume of a cylinder is π × r² × h: 3 × 3 = 9, 9 × 7 = 63, and 3.14 × 63 = 197.82.
- 29.22. Read the two rows: Tuesday 51, Thursday 29. Subtract: 51 − 29 = 22.
- 30.11. Add them: 12 + 7 + 9 + 16 = 44. Divide by how many there are: 44 ÷ 4 = 11.
- 31.8. Put them in order first: 3, 6, 8, 9, 11. The median is the middle one: 8.
- 32.17. The range is the largest minus the smallest: 22 − 5 = 17.
- 33.14. Probability is the favorable outcomes over all the outcomes. There are 3 blue out of 4 + 3 + 5 = 12 marbles: 312, which reduces to 14.
- 34.70%. Either it rains or it does not, and the two chances together make one whole, which is 100%: 100 − 30 = 70.
- 35.12. Each of the 4 sandwiches can go with any of the 3 soups: 4 × 3 = 12. Counting a choice made in two steps means multiplying the steps.
- 36.64. 4³ means 4 × 4 × 4. 4 × 4 = 16, and 16 × 4 = 64.
- 37.9. The square root of 81 is the number that, multiplied by itself, gives 81: 9 × 9 = 81.
- 38.x8. Same base, so keep the base and add the exponents: 5 + 3 = 8. Five factors of x times three more makes eight factors of x.
- 39.1. Any number except 0 raised to the power 0 is 1. One way to see it: 5³ ÷ 5³ = 1, and the subtract-the-exponents rule says that same division is 50.
- 40.6.2 × 106. Move the decimal point until one digit stands before it: 6.2. It moved 6 places, so the power of ten is 6.
- 41.3x + 10. Gather like with like. The x terms: 5x − 2x = 3x. The plain numbers: 3 + 7 = 10.
- 42.8x − 20. Multiply each term inside by 4: 4 × 2x = 8x, and 4 × (−5) = −20.
- 43.x² + x − 6. Multiply every term in the first binomial by every term in the second: x × x = x², x × (−2) = −2x, 3 × x = 3x, and 3 × (−2) = −6. Gather the x terms: −2x + 3x = x.
- 44.x = 5. Undo the addition first: take 7 from both sides, 3x = 15. Then undo the multiplication: divide both sides by 3, x = 5. Check: 3 × 5 + 7 = 22.
- 45.x = 6. Gather the x terms on one side: take 2x from both sides, 3x − 4 = 14. Add 4 to both sides: 3x = 18. Divide by 3: x = 6.
- 46.x > 6. Solve it as you would an equation: add 3 to both sides, 2x > 12, then divide both sides by 2, x > 6. The sign stays as it is, because you divided by a positive number.
- 47.5 hours. Let h be the hours: 60 + 45h = 285. Take 60 from both sides: 45h = 225. Divide by 45: h = 5.
- 48.2. Slope is rise over run. Rise: 11 − 3 = 8. Run: 6 − 2 = 4. Slope = 8 ÷ 4 = 2.
- 49.Slope −3, y-intercept 5. In y = mx + b, the number multiplying x is the slope and the number added is the y-intercept, where the line crosses the y-axis.
- 50.2. Read the marked points: (0, 1) and (2, 5). Rise: 5 − 1 = 4. Run: 2 − 0 = 2. Slope = 4 ÷ 2 = 2.
- 51.x = 8, y = 4. Add the two equations and the y terms cancel: 2x = 16, so x = 8. Put x back into the first: 8 + y = 12, so y = 4.
- 52.17. f(5) means put 5 in place of x: 4 × 5 − 3 = 20 − 3 = 17.
- 53.x = 8. Set the rule equal to 25 and solve: 2x + 9 = 25, so 2x = 16 and x = 8.
- 54.(4, 3). Start at 0. Go 4 to the right, then 3 up. The first number is the distance across, the second is the distance up.
- 55.(2, 4). The midpoint averages the x values and averages the y values: (−2 + 6) ÷ 2 = 2, and (1 + 7) ÷ 2 = 4.