Division S Sample Answers
A. - 35 (apples)
METHOD 1: Examine the common multiples of 3 and 4.
- The least common denominator of 1/3 and 1/4 in 12. Only multiples of 12 will be divisible by 3 and 4. The actual number of whole apples therefore is a multipe of 12. Suppose, for the moment, that the total number of apples picked is 12. Then Jenny would pick 3 apples, Lenny would pick 4 apples, and the difference would be 1.
However, it is given that the actualy difference is 7. Therefore, multiply all quantities by 7: the trio picks a total of 84 apples, Jenny picks 21 apples, Lenny picks 28 apples, and the two together pick a total of 49 apples. Since Penny picks the rest, Penny picks 84 - 49 = 35 apples.
METHOD 2: Use algebra.
- Let x = the total number of apples picked.
Then Jenny picks 1/3 x apples and Lenny picks 1/4 x apples.
1/3x - 1/4x = 7
4/12x - 3/12x = 7
1/12x = 7
Multiply both sides of the equation to get x = 84. Therefore, the total number of apples picked is 84, Jenny picks 1/3 of 84 = 28 apples, Lenny picks 1/4 + of 84 = 21 apples, and Penny picks 84 - (28 + 21) = 35 apples.
METHOD 3: Consider each person's fraction of the total.
- Jenny and Lenny together pick 1/4 + 1/3 = 7/12 of the apples. Thus Penny picks the remaining 5/12 of the apples. Lenny picks 1/3 - 1/4 = 1/12 more of the apples than Jenny, which represents 7 apples. So the total number of apples picked is 12 x 7 = 84. Then Penny picsk 5/12 of 84 = 35 apples.
Let x = the total number of apples picked.
Then Jenny picks 1/3x apples and Lenny picks 1/4x apples.
1/3x - 1/4x = 7
4/12x - 3/12x = 7
1/12x = 7
Multiply both sides of the equation to get x = 84. Therefore, the total number of apples picked is 84, Jenny picks 1/3 of 84 = 28 apples, Lenny picks 1/4 + of 84 = 21 apples, and Penny picks 84 - (28 + 21) = 35 apples.
B. - 960 (dollars)
Track the percent changes...
- $1200 is 120% of the $1000. Increasing an amount by 20% is equivalent to finding 120% of that amount, so that the October price is 120% of the September price. Thus, 1.20 x 1000 = $1200. Simliarly, a decrease of 20% is equivalent to finding 80% of the amount. Thus, 80% of 1200 - $960.
Alternately, replace the two percents with a single percent. Thus, 80% of 120% = 96%. Then 96% of $1000 is $960.
C. - 4 (questions)
Consider the fewest number of correct answers; check even vs. odd.
- To score 59 points one needs to get at least 12 questions right (5 x 12 = 60). Because 59 is odd but the total deduction (at 2 points each) is even, then the number of correct answers (at 5 points each) is odd. Thus Jana has 13, 15, 17, or 19 correct answers. Suppose Jana has 13 correct answers: 5 x 13 = 65. Then the deduction would be 65 - 59 = 6 points for 3 incorrect answers. Therefore, Jana omitted 20 - 13 - 3 = 4 questions.
Do any other answers produce a score of 59?? For every two additional correct answers (10 additional points), Jana needs 5 additional incorrect answers (10 points less). Then these seven additional questions produce a total of at least twenty-three questions. Since only 20 questions are asked, just one way exists to score 59 points.
D. - 900 (palindromes)
Examine the digits from left to right...
- The form of the palindrome is ABCBA. Since 0 is never a leading digit, A (on the left) can be any digit from 1 through 9, and B(on the left) can be any digit from 0 through 9. For each of the 9 possible values of A, there are 10 poissble values of B, a total of 9 x 10 = 90 different two-digit numbers AB. Similarly, C can be any digit from 0 through 9.
For each of the 90 possible values of AB, there are 10 possible values of C, a total of 90 x 10 = 900 different values of the three-digit ABC. Since B and A (on the right) each have only 1 possible value, there is a total of 9 x 10 x 10 x 1 x 1 = 900 five-digit palindromes.
E. - 84 (tiles)
Count tiles
- The length of one side of each tile is 3/4 of a foot. Thus the length of the floor is 18 feet divided by 3/4 foot = 24 tiles, and the width of the floor is 15 divided by 3/4 = 20 tiles.
METHOD 1A
Counting tiles along the border produces 24 = 20 = 24 = 20 = 88 tiles. However, each of the 4 corner tiles was counted twice. Thus, 84 tiles are white.
METHOD 1B
The shorter two sides of the border contain 20 tiles each. This includes the corner tiles. Thus, the longer two sides contain 24 - 2 = 22 uncounted tiles each. Then the border contains 20 + 20 + 22 + 22 = 84 tiles.
METHOD 2
Convert to tiles and then subtract areas
The entire floor consists of 24 x 20 = 40 tiles. The blue portion of the floor consists of 22 x 18 = 396 tiles. Thus, 480 - 396 = 84 tiles are white.
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