Betty used a calculator to find the product 0.075 × 2.56. She forgot to enter the decimal points. The calculator showed 19200. If Betty had entered the decimal points correctly, the answer would have been
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Answer: B — .192.
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Hint 1 of 2
Count the total number of digits after the decimal point in the two factors.
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Hint 2 of 2
Move the decimal point in 19200 that many places to the left.
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Approach: count decimal places
0.075 has 3 decimal places, 2.56 has 2 — total 5. Move the decimal in 19200 five places to the left.
Suppose the estimated 20 billion dollar cost to send a person to the planet Mars is shared equally by the 250 million people in the U.S. Then each person's share is
There are 8 lines (3 rows, 3 columns, 2 diagonals); each must miss at least one square.
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Hint 2 of 2
Leaving the center plus two opposite corners empty kills every line.
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Approach: find a smallest set of empty squares that breaks every line
Each of the 8 lines (3 rows, 3 cols, 2 diagonals) must contain at least one empty square. A single empty square covers at most 4 of those lines (only the center does that much), so 2 empties cover at most 4 + 3 = 7 lines — not enough. So at least 3 squares must be empty.
Three is achievable: leave the center and two opposite corners empty. Center kills the middle row, middle column, and both diagonals; the two opposite corners kill the four remaining edge lines. So 9 − 3 = 6 X's can be placed.
A fifth number, n, is added to the set {3, 6, 9, 10} to make the mean of the set of five numbers equal to its median. The number of possible values of n is
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Answer: C — 3.
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Hint 1 of 2
The median depends on where n falls in the sorted list; split into three cases.
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Hint 2 of 2
In each case, set mean = median and solve for n.
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Approach: case on where n sits in the sorted list
Mean = (28 + n)⁄5. Case n ≤ 6: median = 6 → n = 2. Case 6 ≤ n ≤ 9: median = n → 28 + n = 5n → n = 7. Case n ≥ 9: median = 9 → n = 17.
Each value of n is consistent with its case, so the answers are 2, 7, 17 — 3 values.
Tom's Hat Shoppe increased all original prices by 25%. Now the shoppe is having a sale where all prices are 20% off these increased prices. Which statement best describes the sale price of an item?
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Answer: E — The sale price is the same as the original price.
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Hint 1 of 2
Apply each percent change as a multiplier and multiply them.
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Hint 2 of 2
1.25 × 0.80 — what does that equal?
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Approach: multiply the two factors
+25% means ×1.25; −20% off means ×0.80. Combined: 1.25 × 0.80 = 1.00.
Maria buys computer disks at a price of 4 for $5 and sells them at a price of 3 for $5. How many computer disks must she sell in order to make a profit of $100?
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Answer: D — 240.
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Hint 1 of 2
Cost per disk = $5⁄4; sale price per disk = $5⁄3.
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Hint 2 of 2
Profit per disk = $5⁄3 − $5⁄4.
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Approach: profit per disk × number = total profit
Profit per disk = 5⁄3 − 5⁄4 = 20⁄12 − 15⁄12 = 5⁄12 dollar.
To make $100: 100 ÷ (5⁄12) = 100 × 12⁄5 = 240 disks.
When a square rolls around a hexagon's corner, it pivots through the hexagon's exterior angle.
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Hint 2 of 2
Total rotation from position 1 to position 4 = 3 × 60° = 180°.
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Approach: count pivots × exterior-angle rotation
Each pivot at a hexagon corner rotates the square by the exterior angle 60°. Three pivots take it from the top to the bottom, so the square has rotated 3 × 60° = 180° clockwise.
Rotating the original triangle by 180° gives the orientation in choice A.
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: 1:01, 4:44, and 12:21. How many times during a 12-hour period will be palindromes?
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Answer: A — 57.
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Hint 1 of 2
Split into 1-digit hours (1–9) and 2-digit hours (10–12) and count each.
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Hint 2 of 2
For h:mm to be a palindrome, the last digit of mm must equal h, and the middle digit can be 0–5.
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Approach: split by hour digit count
1-digit hours h:mm (h = 1–9): need mm's ones digit = h with mm's tens digit 0–5; that's 6 minutes per hour, so 9 × 6 = 54 palindromes.
2-digit hours hh:mm (10, 11, 12): only 10:01, 11:11, 12:21 work — 3 more. Total = 54 + 3 = 57.
