Algebraic Shortcuts
If (x + 3)2 - (x - 3)2 = 72, find x.
How many diagonals does a decagon have?
35
The sixth decile is equivalent to what percentile?
60th percentile
How many positive divisors does 72 have?
12
The ratio of boys to girls in a club is 4:5. After 6 additional boys join, the numbers of boys and girls become equal. How many students were originally in the club?
54
If a + b = 12 and ab = 32, find a2 + b2.
80
The bases of a trapezoid measure 11 cm and 19 cm. Find the length of its midsegment.
15 cm
Events A and B are are mutually exclusive. If P(A) = 0.42 and P(B) = 0.31, find P(A U B).
0.73
Find the digit d such that 47d2 is divisible by 9.
5
A map has scale 1:250,000. Two towns are 7.2 cm apart on the map. Find their actual distance in kilometers.
18 km
Let f(x) = x2 - 4x + 7. Simplify f(x + 1) - f(x).
2x - 3
One angle of a cyclic quadrilateral measures 123 degrees. Find the measure of its opposite angle.
57 degrees
A four-digit security code is formed from the digits 0 through 9. Digits cannot repeat, and the first digit cannot be 0. How many different codes are possible?
4536
What is the smallest positive integer that must be multiplied by 180 to make the product a perfect square?
5
A father's present age is three times his son's present age. Four years ago, the father was five times as old as his son. Find the father's present age.
24
2x+1 = 8x-1. Solve for x.
x = 2
A right triangle has legs measuring 9 cm and 12 cm. Find the length of the median drawn from the right-angle vertex to the hypotenuse.
7.5 cm
Three fair six-sided dice are rolled. What is the probability that exactly one die shows a 6?
9936
Find the greatest four-digit number divisible by 72.
9936
The edge length of a cube is increased by 20%. By what percent does the cube's surface area increase?
44%
If x + y = 7 and x2 + y2 = 29, find x3 + y3.
133
Two secants intersect outside a circle. The angle formed by the secants measures 38 degrees. The larger intercepted arc measures 164 degrees. Find the measure of the smaller intercepted arc.
88 degrees
Five fair coins are tossed. What is the probability of obtaining at least four heads?
3/16
How many trailing zeros are there in 75!?
18
A rectangular garden measures 20 m x 12 m. A uniform path is constructed outside the garden on all four sides. If the combined area of the garden and path is 384 m2, find the width of the path.
2 m