Line up the decimal points and give every number the same number of decimal places by adding zeros on the right.
Step-by-step solution
Write each number with 4 decimal places so they line up: 0.6000, 0.5900, 0.1250, 0.0999.
Compare the tenths digit first (the first digit after the point): 6, 5, 1, 0.
The largest tenths digit is 6, so 0.6 is the greatest. No need to look further.
Full order from least to greatest: 0.0999 < 0.125 < 0.59 < 0.6.
The greatest number is 0.6.
Check:0.6 = 0.60 and 0.60 > 0.59 because 60 hundredths is more than 59 hundredths. ✓
Tip: More digits does not mean bigger. Pad with zeros so every decimal has the same length, then compare digit by digit from left to right.
Question 2 of 36
No calculatorHot spot
Click the point on the number line that represents \dfrac{5}{4}.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
First figure out how much each small space is worth: how many spaces are between 0 and 1?
Step-by-step solution
Find the size of one space. Between 0 and 1 there are 4 equal spaces, so each space is \tfrac{1}{4}.
Rewrite the fraction as a mixed number: \dfrac{5}{4} = \dfrac{4}{4} + \dfrac{1}{4} = 1\tfrac{1}{4}.
Start at 0 and count 5 spaces to the right: \tfrac14, \tfrac24, \tfrac34, \tfrac44 = 1, \tfrac54.
That lands on the first tick mark after 1. As a decimal, \tfrac{5}{4} = 1.25.
Answer: the first tick mark to the right of 1 (the point 1.25).
Check:1.25 is between 1 and 1.5, and 1.5 is the middle of 1 and 2. The point is a quarter of the way from 1 to 2. ✓
Tip: On any number line, find the value of one space first: (difference between two labels) \div (number of spaces between them). Count spaces, not tick marks.
Question 3 of 36
No calculator
Which list shows the numbers in order from least to greatest?
\frac{2}{3},\quad 0.6,\quad \frac{5}{8}
Hint
Turn each fraction into a decimal (divide the top by the bottom), then compare.
Step-by-step solution
Convert each fraction to a decimal (3 places is enough):
Tip: Memorize the common ones: \tfrac18 = 0.125, \tfrac14 = 0.25, \tfrac38 = 0.375, \tfrac12 = 0.5, \tfrac58 = 0.625, \tfrac23 \approx 0.667, \tfrac34 = 0.75. They make ordering fast without a calculator.
Question 4 of 36
No calculator
A pharmacy technician weighs four tablets. Their masses, in grams, are shown.
Tablet
Mass (g)
A
0.35
B
0.305
C
0.3
D
0.351
Which tablet is the heaviest?
Hint
Give every mass the same number of decimal places by adding zeros at the end, then compare digit by digit from the left.
Step-by-step solution
Write every mass with 3 decimal places: 0.350, 0.305, 0.300, 0.351.
All the tenths digits are 3, so look at the hundredths digits: 5, 0, 0, 5. Tablets A and D are still tied with 5.
Compare A and D at the thousandths place: 0 vs 1. D has the larger digit.
Full order from least to greatest: 0.300 < 0.305 < 0.350 < 0.351.
The heaviest is Tablet D (0.351 g).
Check:0.351 - 0.350 = 0.001 > 0, so D is heavier than A by one thousandth of a gram. ✓
Tip: Do not stop comparing when two numbers look tied. 0.35 and 0.351 match for two places; the third place decides it.
Question 5 of 36
No calculator
Three hikers are walking the same trail. Mia has finished \dfrac{4}{5} of it, Jon has finished 0.75 of it, and Rosa has finished \dfrac{7}{9} of it.
Which list orders the hikers from farthest along to least far along?
Hint
Change both fractions to decimals by dividing the top number by the bottom number.
Write all four sizes in the same form — either all as decimals, or all as fractions with denominator 32.
Step-by-step solution
Change every fraction to 32nds (or to a decimal):
\dfrac{3}{8} = \dfrac{12}{32} = 0.375
\dfrac{5}{16} = \dfrac{10}{32} = 0.3125
\dfrac{11}{32} = 0.34375
0.4 = \dfrac{12.8}{32}, or simply 0.4
List the decimals: 0.375,\ 0.3125,\ 0.4,\ 0.34375.
Order them: 0.3125 < 0.34375 < 0.375 < 0.4.
The smallest is 0.3125, which is \tfrac{5}{16}.
Answer: the \mathbf{\tfrac{5}{16}}-inch bolt.
Check:\tfrac{5}{16} = \tfrac{10}{32} and \tfrac{11}{32} is the next size up; 10 < 11. ✓
Tip: Bolts, drill bits and wrenches are sized in 8ths, 16ths and 32nds. Turning every fraction into 32nds lets you compare numerators directly.
Question 8 of 36
No calculatorFill in
What number does point P represent? Write your answer as a decimal.
P =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
How many equal spaces are between 2 and 3? Each space is worth 1 divided by that number.
Step-by-step solution
The distance from 2 to 3 is 3 - 2 = 1.
Count the spaces (not the tick marks) between 2 and 3: there are 5 equal spaces.
Size of one space: 1 \div 5 = \dfrac{1}{5} = 0.2.
P is 4 spaces to the right of 2: 2 + 4 \times 0.2 = 2 + 0.8 = 2.8.
Answer: P = 2.8 (which is also 2\tfrac{4}{5}).
Check: one more space would give 2.8 + 0.2 = 3.0, which matches the label 3. ✓
Tip: A common slip is to count tick marks and decide each space is \tfrac14 (giving 3.0 or 2.75 for nearby ticks), or to read 4 spaces as 2.4. Always compute (difference) \div (number of spaces) first.
Question 9 of 36
No calculatorDrag and drop
Drag the numbers into the boxes so they are in order from least (left) to greatest (right).
Least
Greatest
Tap a tile, then tap a box to place it.
Hint
Negative numbers are always less than positive ones. Among negatives, the one farther from zero is smaller.
Check: on a number line, -\tfrac34 is the leftmost point and \tfrac13 the rightmost. ✓
Tip: With negatives, the order flips: 0.75 > 0.5, but -0.75 < -0.5. Picture owing money — owing $0.75 is worse than owing $0.50.
Question 10 of 36
No calculator
Four runners finished a 100-meter race with these times, in seconds:
Runner
Time (s)
Jada
12.09
Luis
12.1
Kim
12.009
Ray
12.19
The runner with the shortest time wins. What was the winning time?
Hint
Shortest time means the smallest number. Write every time with three decimal places.
Step-by-step solution
Give every time three decimal places: 12.090,\ 12.100,\ 12.009,\ 12.190.
The whole-number parts are all 12, so compare the decimal parts.
Tenths digits: 0, 1, 0, 1. The smallest tenths digit is 0 — Jada (12.090) and Kim (12.009).
Compare those two at the hundredths place: 9 vs 0. Kim's 0 is smaller, so 12.009 < 12.090.
Full order: 12.009 < 12.09 < 12.1 < 12.19.
The winning time is 12.009 seconds (Kim).
Check:12.09 - 12.009 = 0.081 > 0, so Kim was faster than Jada. ✓
Tip: Zeros right after the decimal point do matter (0.009 is much less than 0.09). Zeros at the very end do not (12.1 = 12.10).
Question 11 of 36
No calculatorSelect all
Select all numbers that are between\dfrac{1}{2} and \dfrac{3}{4} on the number line.
Select all that apply.
Hint
Change the endpoints to decimals first: what are \tfrac12 and \tfrac34 as decimals?
Step-by-step solution
Convert the endpoints: \tfrac{1}{2} = 0.5 and \tfrac{3}{4} = 0.75. We want numbers with 0.5 < x < 0.75.
Convert each choice:
0.6 → 0.6: between ✓
0.45 → less than 0.5 ✗
\tfrac58 = 0.625: between ✓
0.8 → more than 0.75 ✗
\tfrac23 \approx 0.667: between ✓
0.5 → equal to the left endpoint, so not between ✗
The numbers between are 0.6, \tfrac58, and \tfrac23.
Answer: 0.6, 5/8, and 2/3.
Check:0.5 < 0.6 < 0.625 < 0.667 < 0.75. ✓
Tip: "Between" means strictly inside — the endpoints themselves are not included unless the problem says "between ... inclusive".
Question 12 of 36
No calculator
On a winter morning the temperature was -\dfrac{5}{4} °F. Which point on the number line best represents this temperature?
Hint
Is the number positive or negative? Then change \tfrac54 to a mixed number or a decimal.
Step-by-step solution
The number is negative, so the point must be left of 0. That rules out point A.
Change to a decimal: \dfrac{5}{4} = 5 \div 4 = 1.25, so -\dfrac54 = -1.25.
-1.25 is between -1 and -2: start at -1 and move one more quarter-space (0.25) to the left.
Point C is exactly one tick mark left of -1, at -1.25.
Answer: point C.
Check:-1.25 is less than -1, so it must be to the left of the -1 label — only C is there. ✓
Tip: A fraction whose top is bigger than its bottom (an improper fraction) is more than 1 away from zero. Always check sign first, then size.
Question 13 of 36
No calculatorDrag and drop
Overnight low temperatures, in °F, were recorded in four towns: -1.2, \ -1\tfrac{1}{4}, \ -0.9, and -1.02.
Drag the temperatures into the boxes from coldest (left) to warmest (right).
Coldest
Warmest
Tap a tile, then tap a box to place it.
Hint
Coldest means the smallest number. For negatives, the number farther from zero is smaller.
Step-by-step solution
Write each temperature as a decimal with 2 places: -1.20, \ -1\tfrac14 = -1.25, \ -0.90, \ -1.02.
Think of distance from zero: 1.25, 1.20, 1.02, 0.90.
With negatives, the larger distance means the colder (smaller) temperature. So the order from coldest is -1.25, -1.20, -1.02, -0.90.
Replace with the original forms: -1\tfrac14,\ -1.2,\ -1.02,\ -0.9.
Answer:-1\tfrac14 < -1.2 < -1.02 < -0.9.
Check:-0.9 is the only value between -1 and 0, so it must be the warmest; -1\tfrac14 is farthest left of -1, so it is the coldest. ✓
Tip:-1.02 and -1.2 are easy to mix up. Pad with zeros (-1.02 vs -1.20) and remember the order flips for negatives: 1.20 > 1.02, so -1.20 < -1.02.
Question 14 of 36
No calculatorFill in
A lab gauge is shown as a number line. What value does point Q show? Write your answer as a decimal.
Q =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The labels are not 1 apart. Find the difference between the labels, then divide by the number of spaces between them.
Step-by-step solution
Difference between the labels: 1.6 - 1.4 = 0.2.
Count the spaces between 1.4 and 1.6: there are 4 equal spaces.
Size of one space: 0.2 \div 4 = 0.05.
Q is 3 spaces to the right of 1.4: 1.4 + 3 \times 0.05 = 1.4 + 0.15 = 1.55.
Answer: Q = 1.55.
Check: one more space gives 1.55 + 0.05 = 1.60, which matches the label 1.6. ✓
Tip: Do not assume each space is 0.1. That guess gives 1.4 + 0.3 = 1.7, which is past the 1.6 label — a sure sign of a mistake. Always compute (difference) \div (number of spaces).
Question 15 of 36
No calculator
Four towns measured rainfall, in inches, during one storm.
Town
Rainfall (in)
Ashby
\tfrac{5}{8}
Brook
0.6
Clayton
0.65
Dover
\tfrac{11}{16}
Which town got the most rain?
Hint
Write every amount as a decimal with 4 places, or write every amount in sixteenths.
Step-by-step solution
Convert the fractions to decimals:
Ashby: \dfrac{5}{8} = 5 \div 8 = 0.625
Dover: \dfrac{11}{16} = 11 \div 16 = 0.6875
Line up all four with 4 decimal places: 0.6250, 0.6000, 0.6500, 0.6875.
Order from greatest to least: 0.6875 > 0.6500 > 0.6250 > 0.6000.
The greatest is 0.6875 inches, in Dover.
Answer: Dover.
Check (sixteenths):\tfrac58 = \tfrac{10}{16}, and \tfrac{11}{16} > \tfrac{10}{16}. Also 0.65 = 10.4 sixteenths, still less than 11. ✓
Tip: A bigger denominator means smaller pieces, but you must also count how many pieces there are. Convert first, then compare.
Question 16 of 36
A factory checks items from four production lines.
Line
Inspection result
A
9 defective out of 40
B
defect rate 0.2
C
7 defective out of 32
D
11 defective out of 50
Which line has the highest defect rate?
Hint
A defect rate is (number defective) \div (number checked). Write each rate as a decimal with at least 3 places.
Step-by-step solution
Turn every result into a decimal:
A: \dfrac{9}{40} = 9 \div 40 = 0.225
B: 0.2 = 0.200
C: \dfrac{7}{32} = 7 \div 32 = 0.21875
D: \dfrac{11}{50} = 11 \div 50 = 0.22
Write with 4 places: 0.2250, 0.2000, 0.2188 (rounded), 0.2200.
Order from highest: 0.2250 > 0.2200 > 0.2188 > 0.2000.
Answer: Line A has the highest defect rate.
Check:\tfrac{9}{40} = \tfrac{22.5}{100} and \tfrac{11}{50} = \tfrac{22}{100}, so A is half a percentage point higher than D. ✓
Tip: The line with the most defective items (D, with 11) is not automatically the worst — it also checked more items. Always divide before you compare.
Question 17 of 36
No calculatorSelect all
A store's stock price changed by these amounts, in dollars, on seven different days. A day counts as "quiet" if the change was strictly greater than -\dfrac{1}{2} dollar and strictly less than \dfrac{1}{4} dollar.
Which changes were quiet days? Choose all that apply.
Select all that apply.
Hint
Change the endpoints to decimals first. Remember that a negative number farther from zero is smaller.
Step-by-step solution
Convert the endpoints: -\tfrac12 = -0.5 and \tfrac14 = 0.25. We need -0.5 < x < 0.25.
Check each choice:
-0.6: farther from zero than -0.5, so -0.6 < -0.5 ✗
Tip: Sketch a quick number line with the two endpoints. Anything left of -0.5 or right of 0.25 is out, and the endpoints themselves are out.
Question 18 of 36
No calculatorDrop-down
While checking a contractor's measurements, an inspector compares fractions with the decimals on a digital tape. Use the drop-down menus to choose <, >, or = for each comparison.
\dfrac{5}{6}0.83
-\dfrac{2}{7}-0.3
Choose an answer from each menu.
Hint
Divide to at least 4 decimal places. For the negative pair, compare the distances from zero first, then flip.
Step-by-step solution
Statement 1
\dfrac{5}{6} = 5 \div 6 = 0.8333\ldots
Compare with 0.83 = 0.8300. They match up to the hundredths; at the thousandths place, 3 > 0.
Tip: Rounding \tfrac56 to 0.83 hides the difference. Keep extra decimal places, and with negatives flip the comparison of the sizes.
Question 19 of 36
Fill in
Two friends are on the same straight bike path. Dana is at mile marker 2\tfrac{3}{4}, and Eli is at mile marker 4.1. They agree to meet exactly halfway between their positions.
At what mile marker will they meet? Write your answer as a decimal.
Mile marker
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The halfway point is the average of the two positions: add them, then divide by 2.
Step-by-step solution
Write both positions as decimals: 2\tfrac34 = 2.75 and 4.1 = 4.10.
Add them: 2.75 + 4.10 = 6.85.
Divide by 2: 6.85 \div 2 = 3.425.
Answer: mile marker 3.425.
Check: distance Dana rides: 3.425 - 2.75 = 0.675. Distance Eli rides: 4.1 - 3.425 = 0.675. Equal, so it is the halfway point. ✓
Tip: Two common slips: stopping after adding (6.85 is past both friends — impossible), or taking half of the distance (1.35 \div 2 = 0.675) and forgetting to add it to Dana's starting point.
Question 20 of 36
No calculator
Keisha says, "0.45 is greater than 0.5 because 45 is greater than 5."
Which statement best explains Keisha's mistake?
Hint
Give both decimals the same number of decimal places, then compare.
Step-by-step solution
Keisha compared 45 and 5 as if they were whole numbers. Decimals do not work that way.
Give both numbers two decimal places: 0.5 = 0.50.
Now compare 0.45 and 0.50: at the tenths place, 4 < 5.
So 0.45 < 0.5 — the opposite of what Keisha said.
Answer: She compared the decimals as whole numbers; 0.5 = 0.50, which is greater than 0.45.
Check: in money, 0.45 is 45 cents and 0.5 is 50 cents. 50 cents is more. ✓
Tip: Money is a great reality check for decimals: read them as dollars and cents. On the GED, "explain the error" questions reward the answer that names the exact wrong step, not just the right number.
Question 21 of 36
No calculator
What number does point R represent?
Hint
Find the value of one space first. Then decide which labeled number R is closest to and which direction to count.
Step-by-step solution
Between -2 and -1 there are 3 equal spaces, so each space is \tfrac13.
R is one space to the right of -2.
Moving right means adding: -2 + \tfrac13 = -1\tfrac{2}{3}.
Another way: R is two spaces to the left of -1: -1 - \tfrac23 = -1\tfrac23.
Answer: -1\tfrac{2}{3} (about -1.67).
Check:-1\tfrac23 should be between -2 and -1 and closer to -2. R is closer to -2. ✓
Tip: On the negative side, the mixed number -1\tfrac23 means -(1 + \tfrac23). Counting "one third" from -1 toward -2 gives -1\tfrac13, which is the other tick mark.
Question 22 of 36
A coffee shop sells four bags of beans. Their weights, in pounds, are
Which list shows the weights from least to greatest?
Hint
Write every weight as a decimal with 4 places. Fractions with 3 or 7 in the denominator repeat — divide carefully.
Step-by-step solution
Convert each weight (4 decimal places):
1.3 = 1.3000
1\tfrac27 = 1 + 2 \div 7 \approx 1.2857
\tfrac43 = 1.3333\ldots
1.33 = 1.3300
Order: 1.2857 < 1.3000 < 1.3300 < 1.3333.
Replace with the original forms: 1\tfrac27,\ 1.3,\ 1.33,\ \tfrac43.
Answer: 1\tfrac27,\ 1.3,\ 1.33,\ \tfrac43.
Check:\tfrac43 - 1.33 = 1.3333\ldots - 1.33 = 0.0033\ldots > 0, so \tfrac43 really is the largest. ✓
Tip:\tfrac43 and 1.33 look equal if you round to 2 places. On close calls, keep 4 decimal places, and remember \tfrac13 = 0.333\ldots never stops.
Question 23 of 36
A basketball coach compares four players' free-throw records this season:
Player
Record
Ana
made 17 of 25 shots
Ben
success rate 0.7
Cara
made 27 of 40 shots
Dev
success rate 0.67
Which list orders the players from the highest success rate to the lowest?
Hint
A success rate is (shots made) \div (shots taken). Turn every record into a decimal with 3 places.
Step-by-step solution
Turn each record into a decimal:
Ana: \dfrac{17}{25} = 17 \div 25 = 0.680
Ben: 0.7 = 0.700
Cara: \dfrac{27}{40} = 27 \div 40 = 0.675
Dev: 0.67 = 0.670
Order from highest to lowest: 0.700 > 0.680 > 0.675 > 0.670.
Match names: Ben, Ana, Cara, Dev.
Answer: Ben, Ana, Cara, Dev.
Check: Cara vs Dev is the closest call: 0.675 - 0.670 = 0.005 > 0, so Cara is just ahead. And \tfrac{17}{25} = \tfrac{68}{100} = 0.68. ✓
Tip: Do not round too early. Rounding Cara's 0.675 to 0.67 (or to 0.68) would create a false tie. Keep at least one more decimal place than the numbers you compare.
Question 24 of 36
Fill in
On a number line, point M is at -\dfrac{2}{3} and point N is at 1\dfrac{1}{4}.
What number is exactly halfway between M and N? Write your answer as a fraction or as a decimal rounded to the nearest hundredth.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The halfway point is the average of the two numbers: add them, then divide by 2. Use a common denominator.
Step-by-step solution
Write both numbers as fractions with denominator 12:
Divide by 2 to find the middle: \dfrac{7}{12} \div 2 = \dfrac{7}{12} \times \dfrac{1}{2} = \dfrac{7}{24}.
As a decimal: 7 \div 24 = 0.291\overline{6} \approx 0.29.
Answer: \dfrac{7}{24} \approx 0.29.
Check: distance from M: \tfrac{7}{24} - (-\tfrac{16}{24}) = \tfrac{23}{24}. Distance to N: \tfrac{30}{24} - \tfrac{7}{24} = \tfrac{23}{24}. Equal, so it is the midpoint. ✓
Tip: Keep the negative sign. Dropping it gives \tfrac{8}{12} + \tfrac{15}{12} = \tfrac{23}{12} and a wrong midpoint of \tfrac{23}{24}. The other classic slip is forgetting to divide by 2, which gives \tfrac{7}{12}.
Question 25 of 36
A park ranger records the weekly change in a lake's water level, in feet:
Week
1
2
3
4
Change (ft)
-0.42
-\tfrac{3}{8}
-0.4
-\tfrac{7}{20}
Which list orders the changes from least to greatest?
Hint
Convert the fractions to decimals. For negatives, the number farther from zero is the least.
Step-by-step solution
Convert to decimals with 3 places: -0.420, \ -\tfrac38 = -0.375, \ -0.400, \ -\tfrac{7}{20} = -0.350.
Sizes (distance from zero): 0.420, 0.400, 0.375, 0.350.
For negatives, the biggest size is the least number. Least to greatest: -0.420 < -0.400 < -0.375 < -0.350.
Original forms: -0.42,\ -0.4,\ -\tfrac38,\ -\tfrac{7}{20}.
Check: Week 1 had the biggest drop (0.42 ft), so it must be the least change; Week 4 dropped least, so it is the greatest. ✓
Tip: Read negatives as "drops": the biggest drop is the smallest number. Convert every value first, then sort the sizes and reverse.
Question 26 of 36
No calculator
After a bank fee, Carlos's account balance was -\$0.65. Which point on the number line best represents -0.65?
Hint
Check the sign first. Then find -0.6 and -0.7 on the line — where is the number halfway between them?
Step-by-step solution
The number is negative, so the point is left of 0. That rules out D.
-0.65 is between -0.6 and -0.7 (it is 0.65 away from zero, more than 0.6 but less than 0.7).
It is exactly halfway between those two tick marks.
Point B sits halfway between the ticks for -0.6 and -0.7.
Answer: point B.
Check:-0.65 should be a little more than halfway from 0 to -1 (since 0.65 > 0.5). B is just past the -0.5 label toward -1. ✓
Tip: On the negative side, count left from zero. Counting 0.65 from the -1 end toward zero lands at -0.35 — a common trap.
Question 27 of 36
Fill in
A fuel gauge is drawn as a number line from empty (0) to half full (\tfrac12). What fraction of a full tank does point T show?
Write your answer as a fraction, or as a decimal rounded to the nearest hundredth.
T =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The whole line is only \tfrac12 long. Divide \tfrac12 by the number of spaces to get the size of one space.
Step-by-step solution
The line goes from 0 to \tfrac12, so its length is \tfrac12.
Count the spaces between 0 and \tfrac12: there are 6 equal spaces.
Size of one space: \dfrac12 \div 6 = \dfrac12 \times \dfrac16 = \dfrac{1}{12}.
T is 5 spaces from 0: 5 \times \dfrac{1}{12} = \dfrac{5}{12}.
As a decimal: 5 \div 12 = 0.41\overline{6} \approx 0.42.
Answer: T = \dfrac{5}{12} \approx 0.42.
Check: one more space gives \tfrac{6}{12} = \tfrac12, which matches the right-hand label. ✓
Tip: Do not assume the line ends at 1. If you treat each space as \tfrac16, you get \tfrac56 — more than half a tank, past the end of this line.
Question 28 of 36
Luis needs a board that is at least2\tfrac{2}{3} feet long for a shelf. He has scrap boards that are 2.6 ft, 2.65 ft, 2.67 ft, and 2.7 ft long.
To waste as little wood as possible, which is the shortest board that is long enough?
Hint
Write 2\tfrac23 as a decimal with 3 or 4 places. A board works only if it is greater than or equal to that number.
Step-by-step solution
Convert the requirement: \tfrac23 = 0.6666\ldots, so 2\tfrac23 = 2.6666\ldots \approx 2.667 ft.
Check each board against 2.6667:
2.6 = 2.6000 → too short ✗
2.65 = 2.6500 → too short ✗
2.67 = 2.6700 → long enough ✓
2.7 = 2.7000 → long enough ✓
Two boards work. The shorter of them is 2.67 ft.
Answer: the 2.67-ft board.
Check:2.67 - 2.6667 \approx 0.003 ft of extra wood (about 0.04 in), while 2.7 would leave about 0.033 ft extra. ✓
Tip: Questions with "at least" and "shortest" (or "at most" and "largest") have two steps: first throw out the options that fail, then pick the best of the rest.
Question 29 of 36
Fill in
Three softball teams finished the season with these records:
Team
Record
Hawks
won 13 of 20 games
Lions
won 0.62 of their games
Tigers
won 21 of 32 games
What is the winning rate of the team with the best record? Write it as a decimal rounded to the nearest thousandth.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Divide games won by games played for each team. Compare all three before you round.
Step-by-step solution
Find each winning rate:
Hawks: \dfrac{13}{20} = 13 \div 20 = 0.65
Lions: 0.62
Tigers: \dfrac{21}{32} = 21 \div 32 = 0.65625
Compare with 5 decimal places: 0.65000, 0.62000, 0.65625.
The best is the Tigers, 0.65625.
Round to the nearest thousandth: the digit after the thousandths place is 2, so round down: 0.656.
Answer: 0.656 (the Tigers).
Check:0.656 \times 32 \approx 20.99, which is about 21 wins. ✓
Tip: Rounding both \tfrac{13}{20} and \tfrac{21}{32} to two places gives 0.65 and 0.66 — fine here, but compare with full decimals first so close calls never fool you. Then round only the final answer.
Question 30 of 36
Which fraction is between\dfrac{5}{7} and \dfrac{3}{4}?
Hint
Change the two endpoints and every answer choice to decimals with 3 places.
Step-by-step solution
Convert the endpoints: \dfrac57 \approx 0.714 and \dfrac34 = 0.750. We need 0.714 < x < 0.750.
Convert each choice:
\tfrac{8}{11} = 8 \div 11 \approx 0.727 → between ✓
\tfrac68 = 0.750 → equal to \tfrac34 ✗
\tfrac{7}{10} = 0.700 → less than 0.714 ✗
\tfrac{13}{16} = 0.8125 → more than 0.750 ✗
Answer: \dfrac{8}{11}.
Check (cross-multiplying):\tfrac57 vs \tfrac8{11}: 5 \times 11 = 55 < 7 \times 8 = 56, so \tfrac57 < \tfrac8{11}. And \tfrac8{11} vs \tfrac34: 8 \times 4 = 32 < 11 \times 3 = 33, so \tfrac8{11} < \tfrac34. ✓
Tip: Watch for choices that are equal to an endpoint in disguise, like \tfrac68 = \tfrac34. Simplify every fraction before deciding.
Question 31 of 36
Drop-down
Select the symbol that makes each statement true.
\dfrac{7}{9}0.78
-0.45-\dfrac{4}{9}
Choose an answer from each menu.
Hint
Divide carefully and keep at least 4 decimal places — these numbers are very close. Ninths make repeating decimals.
Check:0.1875 \times 16 = 3 exactly. And 0.0909 \times 11 = 0.9999 < 1, so 0.0909 < \tfrac{1}{11}. ✓
Tip: Fractions whose denominators use only 2s and 5s (16 = 2^4) give decimals that stop, so an exact "=" is possible. Denominators like 6 or 11 give repeating decimals, which are never equal to a short decimal.
Question 33 of 36
On a cold morning the temperature rose steadily from -\tfrac12 °F to 2\tfrac12 °F. At what temperature had it risen three-fourths of the way from the starting temperature to the final temperature?
Hint
First find the total change. Then take \tfrac34 of that change and add it to the starting temperature.
Three-fourths of the change: \tfrac34 \times 3 = \tfrac94 = 2.25 °F.
Add this to the starting temperature: -0.5 + 2.25 = 1.75 °F.
Answer: 1.75 °F (which is 1\tfrac34 °F).
Check: the remaining rise is 2.5 - 1.75 = 0.75, which is \tfrac14 of the total change of 3. ✓
Tip: "A fraction of the way from a to b" means a + \text{fraction} \times (b - a). Do not forget the starting point, and subtract a negative carefully: 2.5 - (-0.5) = 3, not 2.
Question 34 of 36
No calculator
Which statement is true?
Hint
Convert each fraction to a decimal, then test each statement. With negatives, the number farther from zero is smaller.
-0.25 < -\tfrac13? \ -\tfrac13 = -0.333\ldots, which is farther left than -0.25, so -0.25 > -\tfrac13. False.
Answer: -\tfrac23 < -0.6.
Check: on a number line, -\tfrac23 sits between -0.7 and -0.6, to the left of -0.6. ✓
Tip: In "which statement is true" questions, test every choice quickly. Writing each fraction as a 3-place decimal first makes the tests almost automatic.
Question 35 of 36
No calculator
Marco was asked to order -0.6, \ -\tfrac{2}{3}, and 0.06 from least to greatest. Here is his work.
Step 1:-\tfrac{2}{3} = -0.667 (rounded).
Step 2: Since 0.667 > 0.6, \ -\tfrac{2}{3} > -0.6.
Step 3:0.06 is the only positive number, so it is the greatest.
Step 4: Order: -0.6,\ -\tfrac{2}{3},\ 0.06.
In which step did Marco make his first error?
Hint
Check each step on its own. Is each statement true by itself?
Step 2: It is true that 0.667 > 0.6 for the sizes. But for negatives the order flips: the number farther from zero is smaller. So -\tfrac23 < -0.6. Marco wrote >. This is the first error.
Step 3:0.06 > 0 and the others are negative, so 0.06 is greatest. Correct.
Step 4: This follows from the wrong Step 2. The correct order is -\tfrac23,\ -0.6,\ 0.06.
Answer: Step 2.
Check: owing $0.67 is worse than owing $0.60, so -0.67 < -0.60. ✓
Tip: In "first error" questions, a later step can be wrong only because of an earlier one. Test the steps in order and stop at the first false statement.
Question 36 of 36
No calculator
Gas stations often post prices with a fraction of a cent. Four stations post these prices per gallon:
Station
Price per gallon
A
\$3.49\tfrac{9}{10}
B
\$3.495
C
\$3.48\tfrac{9}{10}
D
\$3.49
Which station has the lowest price?
Hint
The \tfrac{9}{10} is nine-tenths of a cent, and one cent is \$0.01. Write every price with 3 decimal places.
Step-by-step solution
A fraction of a cent: \tfrac{9}{10} of a cent = \tfrac{9}{10} \times \$0.01 = \$0.009.
Write every price with 3 decimal places:
A: \$3.49 + \$0.009 = \$3.499
B: \$3.495
C: \$3.48 + \$0.009 = \$3.489
D: \$3.49 = \$3.490
Compare at the hundredths place: A, B, D have 9; C has 8. So C is the lowest.
Full order: 3.489 < 3.490 < 3.495 < 3.499.
Answer: Station C (\$3.489 per gallon).
Check: C is \$3.490 - \$3.489 = \$0.001 cheaper than D per gallon. ✓
Tip: Price signs like 3.49\tfrac{9}{10} add less than one cent. Converting everything to three decimal places turns a confusing comparison into a simple one.
