FILL BLANKAn oak tree is 20 years older than a maple tree. Fifteen years from now, the oak tree will be twice as old as the maple tree.What is the current age of the oak tree, in years?
Answer(s): A
Let m = current age of the maple tree, o = current age of the oak tree.Given o = m + 20 (oak is 20 years older)Fifteen years from now:o + 15 = 2(m + 15)Substitute o = m + 20:(m + 20) + 15 = 2m + 30m + 35 = 2m + 3035 - 30 = 2m - m5 = mSo, the maple tree is 5 years old, and the oak tree is o = m + 20 = 5 + 20 = 25.
The two-digit positive integer X Y is a multiple of 5, where X and Y are the tens digit and the units digit, respectively.Which of the following statements individually provide(s) sufficient additional information to determine the value of integer X Y? (Choose all that apply.)
Answer(s): A,B,C
A two-digit positive integer XY hat is a multiple of 5 must end in 0 or 5; hence Y {0,5}If X = Y and Y {0,5}, then X = 5 (since a tens digit cannot be 0).Thus, the integer must be 55.X + Y = 10With Y {0,5}If Y=0, then X=10 (impossible for a digit).If Y=5, then X=5.So, the integer must again be 55For a two-digit number, divisibility by 11 means 10X + Y 0 (mod 11) Because 10 -1 (mod 11), this is equivalent to -X + Y 0 (mod 11), i.e. Y = X.Y = X plus Y {0,5} forces X = 5, giving 55.
The research budget at Company X was $3 million in 2015, and the budget increased by 60 percent from 2015 to 2020. The research budget at Company Y was $4 million in 2015, and the budget increased by 45 percent from 2015 to 2020.Quantity AThe increase, in dollars, in the research budget at Company X from 2015 to 2020Quantity BThe increase, in dollars, in the research budget at Company Y from 2015 to 2020
Answer(s): C
Company X:60 % of $3 million = 0.60 × 3= 1.8 million dollarsCompany Y: 0.45 × 4 = 1.8 million dollarsBoth companies' research budgets increased by the same amount. Quantity A = Quantity B
0 < p < 10 < t < 1Quantity AptQuantity B
Answer(s): B
Since p > 0 and t > 0, divide both quantities by p (a positive number) without changing the inequality direction:Hence the comparison reduces to t versus 1/t.Because 0 < t < 1, we have t < 1 < 1/tThus, 1/t > t, so p/t > pt
x < yQuantity AACQuantity B2
Since the triangle's angles satisfy x + y + 45° = 180°, with x < y, we have y > x/In any triangle, the larger interior angle is opposite the longer side:Angle is at , opposite side AB.Angle x is at B, opposite side AC.Because > x, side AB is longer than side AC. But AB is given to be 2. Therefore AC < AB = 2.Quantity A < Quantity B.
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