Math Diagnostic: Flyover II (additional practice)

Family 1

Numbers and the four operations

1.In the number 2,853,719, what is the 5 worth?
2.7,003 − 2,548 =
3.27 × 35 =
4.20 − 2 × (9 − 3) ÷ 4 =

Family 2

Signed Numbers (positives & negatives)

5.−8 + 15 =
6.3 − (−9) =
7.−7 × (−6) =

Family 3

Fractions

8.35 + 14 =
9.23 × 38 =
10.34 ÷ 18 =
11.Which is larger: 23 or 710?

Family 4

Decimals

12.9.2 − 4.65 =
13.0.8 × 0.45 =
14.Write 58 as a decimal.
15.A storage locker costs $23.75 a month. What do twelve months cost?

Family 5

Percents, ratios, and proportion

16.What is 35% of 80?
17.A $60 backpack is marked 30% off. What is the sale price?
18.What is 12 out of 40 as a percent?
19.A car uses 4 gallons of gas to go 130 miles. How far can it go on 10 gallons?

Family 6

Measurement and units

20.How many ounces are in 6 pounds?
21.A cooler holds 5 gallons of water. How many quarts is that?
22.A table is 3 feet 4 inches long. How many inches is that?

Family 7

Geometry

23.A triangle has a base of 12 inches and a height of 7 inches. What is its area?
24.A circle has a radius of 5 cm. What is its area? Use 3.14 for π.
25.A right triangle has legs of 5 and 12. How long is the hypotenuse?
26.Two lines cross. One of the four angles is 48°. What is the angle directly opposite it?
27.A box is 8 inches long, 5 inches wide, and 4 inches tall. What is its volume?
28.A can has a radius of 2 inches and a height of 10 inches. What is its volume? Use 3.14 for π.

Family 8

Data, graphs, and statistics

29.Cups of coffee sold at the cart
Day Cups
Monday 62
Tuesday 45
Wednesday 58
Thursday 71
How many more cups were sold on Thursday than on Tuesday?
30.Find the mean of 15, 9, 13, 11.
31.Find the median of 14, 6, 10, 3, 9.
32.Find the range of 27, 8, 19, 12.

Day	Cups
Monday	62
Tuesday	45
Wednesday	58
Thursday	71

Family 9

Probability and counting

33.A box holds 2 red pens, 6 blue, and 4 black. You take one without looking. What is the probability it is black?
34.The forecast says there is a 45% chance of snow. What is the probability it does not snow?
35.A food cart offers 5 kinds of rice bowl and 4 drinks. How many different bowl-and-drink combinations can be ordered?

Family 10

Exponents, roots, and scientific notation

36.5³ =
37.√144 =
38.Write as a single power: x⁴ × x⁵
39.What is 12⁰?
40.Write 45,000,000 in scientific notation.

Family 11

Expressions and polynomials

41.Simplify: 7x − 4 + 2x + 9
42.Multiply out: 5(3x − 2)
43.Multiply out: (x + 2)(x − 5)

Family 12

Equations and inequalities

44.Solve for x: 4x + 5 = 33
45.Solve for x: 8x − 9 = 5x + 15
46.Solve: 3x + 4 < 19
47.An electrician charges $75 to come out, plus $40 an hour. The bill came to $315. How many hours did the job take?

Family 13

Lines, slope, and systems

48.What is the slope of the line through (1, 2) and (5, 14)?
49.For the line y = 2x − 7, what are the slope and the y-intercept? slope y-intercept
50.The line passes through the two marked points. What is its slope?
51.Solve the system: x + y = 15 and x − y = 3 x y

Family 14

Functions and the coordinate plane

52.If f(x) = 3x + 4, what is f(6)?
53.If f(x) = 5x − 8, for which x is f(x) = 27?
54.What are the coordinates of point B?
55.What is the midpoint of the segment from A (−4, 1) to B (2, 5)?

Or check one family at a time, with the button under it.

Results

Where to go next

Family	Questions	Right	Where to go next
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.

Answer key

Answers, with the reasoning in brief

1.50,000. The 5 stands in the ten-thousands place, so it is worth fifty thousand. Count the places from the right to see it: 9 ones, 1 ten, 7 hundreds, 3 thousands, 5 ten-thousands. The whole number reads: two million, eight hundred fifty-three thousand, seven hundred nineteen.
2.4,455. You cannot take 8 from 3, and the tens and hundreds are zeros, so the borrow comes all the way from the thousands: 7,003 becomes 6 thousands, 9 hundreds, 9 tens, and 13 ones. Then 13 − 8 = 5, 9 − 4 = 5, 9 − 5 = 4, 6 − 2 = 4. Check by adding back: 4,455 + 2,548 = 7,003.
3.945. Break 35 into 30 and 5: 27 × 30 = 810 and 27 × 5 = 135; 810 + 135 = 945. Breaking the 27 instead works just as well: 20 × 35 = 700 and 7 × 35 = 245; 700 + 245 = 945.
4.17. Parentheses first: 9 − 3 = 6. Then multiplying and dividing, left to right: 2 × 6 = 12, and 12 ÷ 4 = 3. Subtracting last: 20 − 3 = 17. Working straight across from the left instead (20 − 2 = 18, and so on) gives a different number, and that is the trap the question sets.
5.7. Start at −8 and go up 15. The first 8 steps bring you to 0; the other 7 take you to 7. Another way to see it: the 15 and the 8 pull in opposite directions, so find the difference, 15 − 8 = 7, and the answer takes the sign of the larger, which is positive.
6.12. Subtracting a negative is the same as adding a positive: 3 − (−9) = 3 + 9 = 12. On a number line, taking away 9 below zero moves you 9 up.
7.42. Two negatives multiplied give a positive: 7 × 6 = 42, and the two minus signs cancel each other. One negative and one positive would give a negative.
8.1720. Fifths and quarters do not match. Twentieths hold both: 35 = 1220 and 14 = 520. Then 12 + 5 = 17 twentieths. Nothing divides both 17 and 20, so it is already in simplest form.
9.14. Multiply straight across: 2 × 3 = 6 on top, 3 × 8 = 24 underneath. 624 reduces to 14. You can also cancel first: the 3 on top and the 3 underneath divide out, leaving 28, which is 14.
10.6. The question is how many eighths fit inside three quarters. Three quarters is six eighths, so the answer is 6. The rule says the same thing: turn the second fraction over and multiply, 34 × 81 = 244 = 6.
11.710. In thirtieths, 23 = 2030 and 710 = 2130. Twenty-one parts beat twenty, so 710 is larger, by one thirtieth. As decimals: 23 is 0.666…, and 710 is 0.7.
12.4.55. Line up the decimal points and write 9.2 as 9.20 so the places match: 9.20 − 4.65 = 4.55. Check: 4.55 + 4.65 = 9.20.
13.0.36. Multiply as whole numbers: 8 × 45 = 360. Then count the decimal places in the question — one and two, three in all — and put three into the answer: 0.360, which is 0.36. It makes sense that the answer is smaller than 0.45: you took less than one whole of it.
14.0.625. A fraction is a division: 5 ÷ 8. Eight does not go into 5, so 5.000 ÷ 8 = 0.625. Or build it from a fact worth knowing: 18 = 0.125, and five of those is 0.625.
15.$285.00. 12 × 23.75: 12 × 23 = 276 and 12 × 0.75 = 9, because three quarters of 12 is 9; 276 + 9 = 285.
16.28. 10% of 80 is 8, so 30% is three of those, 24. 5% is half of 8, which is 4. Together: 24 + 4 = 28. Another way: 35% is 0.35, and 0.35 × 80 = 28.
17.$42. 10% of 60 is 6, so 30% is 18. Take it off: 60 − 18 = 42. Or notice that paying 30% less means paying 70%, and 70% of 60 is 42.
18.30%. Percent means per hundred. 1240 reduces to 310, and three tenths is 30 per hundred. Another way: 40 goes into 100 two and a half times, so 12 × 2.5 = 30.
19.325 miles. One gallon takes the car 130 ÷ 4 = 32.5 miles, so 10 gallons take it 10 × 32.5 = 325 miles. Set up as a proportion it reads 4/130 = 10/x; cross-multiplying gives 4x = 1,300, so x = 325.
20.96. A pound is 16 ounces: 6 × 16 = 96. (6 × 10 = 60 and 6 × 6 = 36; 60 + 36 = 96.)
21.20. A gallon is 4 quarts: 5 × 4 = 20. Going from a larger unit to a smaller one means more of them, so multiply.
22.40. Three feet is 3 × 12 = 36 inches. Add the 4: 36 + 4 = 40.
23.42 square inches. Area of a triangle is half of base × height: 12 × 7 = 84, and half of 84 is 42. A triangle is half of the rectangle that would just hold it, which is where the half comes from.
24.78.5 square cm. Area of a circle is π × r²: 5 × 5 = 25, and 3.14 × 25 = 78.5. The formula is on the sheet; choosing it, and squaring the radius before multiplying by π, is the work.
25.13. Pythagorean theorem: a² + b² = c². 5² + 12² = 25 + 144 = 169, and √169 = 13. The hypotenuse is the longest side, and 13 is longer than both legs, as it should be.
26.48°. When two lines cross, the angles opposite each other are equal. Each pair of neighbors adds to 180°: the angle next to the 48° is 132°, and the angle opposite is 180° − 132° = 48° again, which is what makes it so.
27.160 cubic inches. Volume of a box is length × width × height: 8 × 5 = 40, and 40 × 4 = 160.
28.125.6 cubic inches. Volume of a cylinder is π × r² × h: 2 × 2 = 4, 4 × 10 = 40, and 3.14 × 40 = 125.6. In words: the circle on the bottom has an area of 12.56 square inches, and the can stacks ten of those.
29.26. Read the two rows: Thursday 71, Tuesday 45. Subtract: 71 − 45 = 26.
30.12. Add them: 15 + 9 + 13 + 11 = 48. Divide by how many there are: 48 ÷ 4 = 12.
31.9. Put them in order first: 3, 6, 9, 10, 14. The median is the middle one: 9, with two numbers below it and two above.
32.19. The range is the largest minus the smallest: 27 − 8 = 19.
33.13. Probability is the favorable outcomes over all the outcomes. There are 4 black pens out of 2 + 6 + 4 = 12 pens: 412, which reduces to 13.
34.55%. Either it snows or it does not, and the two chances together make one whole, which is 100%: 100 − 45 = 55.
35.20. Each of the 5 bowls can go with any of the 4 drinks: 5 × 4 = 20. Counting a choice made in two steps means multiplying the steps.
36.125. 5³ means 5 × 5 × 5. 5 × 5 = 25, and 25 × 5 = 125. It is not 5 × 3.
37.12. The square root of 144 is the number that, multiplied by itself, gives 144: 12 × 12 = 144.
38.x⁹. Same base, so keep the base and add the exponents: 4 + 5 = 9. Four factors of x times five more makes nine factors of x.
39.1. Any number except 0 raised to the power 0 is 1. One way to see it: 12² ÷ 12² = 1, and the subtract-the-exponents rule says that same division is 12⁰.
40.4.5 × 10⁷. Move the decimal point until one digit stands before it: 4.5. It moved 7 places, so the power of ten is 7. Check: 4.5 × 10,000,000 = 45,000,000.
41.9x + 5. Gather like with like. The x terms: 7x + 2x = 9x. The plain numbers: −4 + 9 = 5.
42.15x − 10. Multiply each term inside by 5: 5 × 3x = 15x, and 5 × (−2) = −10.
43.x² − 3x − 10. Multiply every term in the first binomial by every term in the second: x × x = x², x × (−5) = −5x, 2 × x = 2x, and 2 × (−5) = −10. Gather the x terms: −5x + 2x = −3x.
44.x = 7. Undo the addition first: take 5 from both sides, 4x = 28. Then undo the multiplication: divide both sides by 4, x = 7. Check: 4 × 7 + 5 = 33.
45.x = 8. Gather the x terms on one side: take 5x from both sides, 3x − 9 = 15. Add 9 to both sides: 3x = 24. Divide by 3: x = 8. Check: 8 × 8 − 9 = 55, and 5 × 8 + 15 = 55.
46.x < 5. Solve it as you would an equation: take 4 from both sides, 3x < 15, then divide both sides by 3, x < 5. The sign stays as it is, because you divided by a positive number. Any number below 5 works: try 4, and 3 × 4 + 4 = 16, which is less than 19.
47.6 hours. Let h be the hours: 75 + 40h = 315. Take 75 from both sides: 40h = 240. Divide by 40: h = 6. Check: 75 + 40 × 6 = 75 + 240 = 315.
48.3. Slope is rise over run. Rise: 14 − 2 = 12. Run: 5 − 1 = 4. Slope = 12 ÷ 4 = 3. The line climbs 3 for every 1 it moves to the right.
49.Slope 2, y-intercept −7. 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. Here the added number is −7, so the line crosses the y-axis below zero, at (0, −7).
50.3. Read the marked points: (1, 2) and (2, 5). Rise: 5 − 2 = 3. Run: 2 − 1 = 1. Slope = 3 ÷ 1 = 3.
51.x = 9, y = 6. Add the two equations and the y terms cancel: 2x = 18, so x = 9. Put x back into the first: 9 + y = 15, so y = 6. Check the second: 9 − 6 = 3.
52.22. f(6) means put 6 in place of x: 3 × 6 + 4 = 18 + 4 = 22.
53.x = 7. Set the rule equal to 27 and solve: 5x − 8 = 27, so 5x = 35 and x = 7. Check: 5 × 7 − 8 = 27.
54.(5, 2). Start at 0. Go 5 to the right, then 2 up. The first number is the distance across, the second is the distance up. Written the other way around, (2, 5) names a different point.
55.(−1, 3). The midpoint averages the x values and averages the y values: (−4 + 2) ÷ 2 = −2 ÷ 2 = −1, and (1 + 5) ÷ 2 = 3. A negative and a positive average the same way as any two numbers: add them, then halve. The picture agrees: the circled point sits one step left of the y-axis and three steps up.