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Question 1 of 10 · Absolute value and distance on a number line
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In golf, a score below par is written as a negative number. At the end of a tournament, Ana's score was -4 (4 under par) and Beth's score was +3 (3 over par).
How many strokes apart were their scores?
Hint
Picture the two scores on a number line with par at 0. How far is -4 from 0? How far is 0 from 3?
Step-by-step solution
Place the scores on a number line: Ana at -4, Beth at 3, par at 0.
Check: from -4 up to par is 4 strokes; from par up to +3 is 3 strokes; 4 + 3 = 7. ✓
Tip: When two values are on opposite sides of zero (under and over par, below and above sea level, owing and having money), add their sizes. Subtracting 4 - 3 = 1 treats both scores as if they were on the same side.
Question 2 of 10 · Order fractions and decimals, including on a number line
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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 3 of 10 · Multiples and factors
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Hot dogs are sold in packages of 10, and hot dog buns are sold in packages of 8. Tasha wants exactly the same number of hot dogs and buns, with none left over.
What is the smallest number of hot dogs she can buy?
Hint
The number of hot dogs must be a multiple of 10, and the number of buns a multiple of 8. You need a number that is on both lists.
Step-by-step solution
Hot dogs come in groups of 10, so possible totals are multiples of 10: 10, 20, 30, 40, 50, \ldots
Buns come in groups of 8: 8, 16, 24, 32, 40, 48, \ldots
The smallest number on both lists is 40, the LCM of 10 and 8.
Prime check: 10 = 2 \times 5, 8 = 2^3, so LCM = 2^3 \times 5 = 40.
She should buy 40 hot dogs (4 packages of hot dogs and 5 packages of buns).
Check:4 \times 10 = 40 and 5 \times 8 = 40. ✓
Tip: Phrases like "the same number", "at the same time again" or "smallest amount to buy" in a word problem point to the LCM. Splitting things into equal groups points to the GCF.
Question 4 of 10 · Multiples and factors
No calculatorSelect all
Select all of the numbers that divide 2{,}340 evenly (with no remainder).
Select all that apply.
Hint
Use the divisibility rules: digit sum for 3 and 9, last two digits for 4 and 25, last three digits for 8.
Step-by-step solution
By 3: digit sum 2 + 3 + 4 + 0 = 9, divisible by 3 ✓
By 4: last two digits 40, and 40 \div 4 = 10 ✓
By 8: last three digits 340, and 340 \div 8 = 42.5 ✗
By 9: digit sum 9, divisible by 9 ✓
By 25: last two digits must be 00, 25, 50 or 75; they are 40 ✗
Answer: 3, 4, and 9.
Check:2340 \div 3 = 780, 2340 \div 4 = 585, 2340 \div 9 = 260. And 2340 = 2^2 \times 3^2 \times 5 \times 13 has only two factors of 2, so 8 = 2^3 cannot divide it. ✓
Tip: Being even does not make a number divisible by 8, and ending in 0 does not make it divisible by 25. Use the full rule for each divisor.
Question 5 of 10 · Multiples and factors
No calculatorDrag and drop
Drag the correct prime factorization under each number. One tile is not used.
36
60
84
Tap a tile, then tap a box to place it.
Hint
Every number in a prime factorization must be prime. Multiply each tile out to see which number it equals.
Tip: Write the subtraction with parentheses when an exponent is negative: -3 - (-4). Dropping the parentheses (-3 - 4 = -7) is the most common error on this kind of item.
Question 8 of 10 · Absolute value and distance on a number line
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In a trivia game, Tom answered one question wrong and lost 350 points. On the next question he gained 200 points. His score is now −75.
What was Tom's score before he answered the question wrong?
points
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Work backwards from the end. Undo the last change first, then undo the one before it.
Step-by-step solution
Write the story as an equation: \text{start} - 350 + 200 = -75.
Undo the gain of 200 (subtract 200): -75 - 200 = -275. This was his score after the wrong answer.
Tip: Work backwards in reverse order and reverse each operation: a gain becomes a subtraction and a loss becomes an addition. A common wrong answer is -225, which applies the changes again (-75 - 350 + 200) instead of reversing them. Always check by running the story forward.
Question 9 of 10 · Order fractions and decimals, including on a number line
Drop-down
For each row, open the menu and pick the comparison sign (<, >, or =) that gives a true statement. Some pairs may be exactly equal.
-1\dfrac{5}{6}-1.83
0.0909\dfrac{1}{11}
\dfrac{3}{16}0.1875
Choose an answer from each menu.
Hint
Divide each fraction to at least 5 decimal places. Some fractions repeat forever, and some stop exactly.
Step-by-step solution
Statement 1
\tfrac56 = 0.8333\ldots, so -1\tfrac56 = -1.8333\ldots
Sizes: 1.8333\ldots > 1.83, so -1\tfrac56 is farther left of zero.
-1\tfrac56 < -1.83.
Statement 2
\tfrac{1}{11} = 1 \div 11 = 0.090909\ldots (the "09" repeats).
Compare with 0.0909 = 0.090900. The first five decimal places match (0.09090). At the sixth place, 0.090900 has 0 and \tfrac1{11} has 9.
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 10 of 10 · Order fractions and decimals, including on a number line
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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.