GRE Math Problem Solving

What is the value of (⁻8)⁻³?

A number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent. Therefore, (⁻8)⁻³=¹/(⁻8)⁻³=¹/(⁻8)(⁻8)(⁻8)=⁻¹/₅₁₂.

If one of the angle measures of an equilateral triangle is given in degrees as 15n, what is the value of n?

equilateral triangle has 3 equal sides. Since the sum of the angles of a triangle always equals 180°, you can divide 180 by 3 to determine the angles (thus 60°). 1. 15n=60 (so take 60/15) 2. n=4

What is the perimeter of a parallelogram with adjacent side lengths measuring 9a & 14b?

the perimeter of the parallelogram is the sum of the lengths of all 4 of its sides. Two sides lengths are given and a parallelogram has 2 pairs of equal opposite sides. so 2(9a) +2(14b)=18a+28b.

If *m=3m and ⊕n=n+7, which of the following is the value of [*(⊕4)-3(⊕2)?

the symbol represents operations on the #'s. To get the answer apply the operation described in the problem statement each time the symbol appears. 1. ⊕n=n+7-->⊕4=4+7=11 & ⊕2=2+7=9. Therefore... 2. [*(11)-3(9)]=[*(11-27)]. Since *m=3m, then.. 3. [3(11)-27]=33-27=6.

Pipe A can fill a tank in 3 hours. If pipe B can fill the same tank in 2 hours, how many minutes will it take both pipes to fill 2/3 of the tank?

1. # of hours it takes to both pipes to fill entire tank x 2/3 ad then convert hours to minutes. 2. The time it takes pipe A & Pipe B to fill the tank (A=3, B=2) is the product of their individual times to do the work (multiply these two terms) divided by the sum of their individual times it takes for their work (A=3, B=2). So (3x2)/(3+2)=6/5. 3. It takes 2 pipes 6/5 hours to fill the entire tank. So take 2/3 x 6/5=2/1x2/5=4/5 4. 4/5 x60=4x12=48 minutes.

A computer Co. featured a laptop cost $800 last year. This year, the laptop sold for 15% less than it did last year. Next year, after updates are made to the model, there will be a 25% price increase over this year's price. What will be the price next year?

1. Calculate the price of the laptop this year ($800 x .15%=120. $800-$120=$680) 2. Use the price on step 1 to determine what the price will be next year. ($680 x 125%=$850). If there is a 25% increase next year, the system would sell for 125% (100% + 25%). ANS=$850.

In company X, no employee is both a technician and an accountant. Also, in company X, ²/₅ of the employees are technicians and ⁵/₁₆ of the remaining employees are accountants. What fraction of the total number of employees at company X are neither technicians nor accountants?

1. Multiply the denominator of ⁵/₁₆ by the reciprocal of ²/₅. (16x5)=80(total # of employees). 2. If ²/₅ is the # of technicians, then ²/₅ x 80=32 technicians. Now if we get 80-32, that leaves us with 48 employees that are not technicians. 3. ⁵/₁₆ is the # of accountants and ⁵/₁₆ x 48=15 (number of accountants). 4. 48-15=33 (the number of employees that are neither technicians or accountants).

Phillip has twice as many tropical fish as Jody. If Phillip gave Jody 10 of his tropical fish, he would have half as many as Jody. How many tropical fish do Phillip and Jody have together?

1. We can assume that the ratio is 2:1 since Phillip has twice as many tropical fish as Jody. Because we have a 2:1 ratio, if we add 2+1=3 then we know that the ANS is a multiple of 3. 2. If Phillip has twice as many fish as Jody, we can write this as p=2j. 3. If Phillip give Jody 10 fish, then he will have 10 fewer and Jody will have 10 more so: p-10=¹/₂(j+10). Now we have two equations to describe the situation. 4. To solve, replace p with j. 5. p-10=¹/₂(j+10)-->2j-10=¹/₂(j+10). Multiply both sides by 2 to collect like terms so that: 2j-10=¹/₂(j+10)-->4j-20=j+10. But do not multiply what is in the parentheses twice. Now solve 6. 4j=j+30-->3j=30 j=10.

The integer y is positive. If 6y is a factor of (2¹⁴) (3²⁴), then what is the greatest possible value of y?

1. rewrite 6y by using prime factorization of 6; thus: 2 x 3. 2. We know by the law of exponents that (ab)n=anbn so: 6y=(2x3)y=2yx3y.

The surface area of a cube with side length (x+4) is 294. What is the value of x?

1st consider a cube has 6 faces and all edges are equal. The surface area is given (294). So divide 294 by 6=49. 2. The area of a square is equal to the square of its side and it is given in the problem (x+4). 3. (x+4)²=49 so (x+4)²=7². We can disregard the negative in this case since distance won't be a negative. Therefore: 4. (x+4=7)-->x=7-4-->x=3.

If it takes Nathan 4 hours to unload a moving truck and it takes Iris 2 hours longer than Nathan to unload a moving truck, how long would it take the two of them, working together to unload 2 moving trucks?

According to the combined work formula, the amount of time it takes two people to do a single task together is the product of their individual times to do the task divided by the sum of those times. It takes Nathan 4 hours and it takes Iris 6 hours (4+2). 1. 4x6/4+6=24/10=12/5. 2. 12/5=2.4 so choose the number closest to this. (2 hours and 24 minutes. 3. the question asks how long it would take both of them to unload TWO moving trucks: ANS: 4 hours and 48 minutes.

The data below shows the monthly dollar amounts Marco spent on postage over the past 6 months. What is the average of the data set? $1.08, $5.43, $2.17, $3.25, $5.95, $1.08.

Average is going to = the sum of terms/# of terms. 18.96/6=$3.16

What is the value of (2√2) (√6) + 2√3?

Following PEMDAS, simplify the first part of the expression. 1. Multiply the radical terms (deal with the inside & outside separately) 2. If the radical terms have the same # under the √, then add them together. 3. (2√2) (√6) + 2√3=(2√2) (√2×√3) + 2√3 4. (2) (√2×√2) (√3) +2√3=4√3 +2√3 5. 4√3+2√3=6√3

What is the value of -[(s+t)⁰] if s +t ≠0?

For any non-zero base raised to the zero power, the value is equal to 1. Since the problem specifies that the base is not zero, it does not matter what s or t is. Their sum raised to the zero power will be 1. The expression asks for a negation of 1 so the answer is -1.

Find the measures, in degrees, of angles GEF and DEF if the angles of GEF=(x+20°) & DEG=(x-60°):

From the general rules about relationships between angles formed by intersecting lines, you know that the sum of the angles along a straight line is 180°. Angles DEG & GEF lie along a straight line. They are supplementary angles. 1. Write an equation using the terms provided: (x+20)+(x-60)=180. Now solve for x. 2. -60+20=-40. X+X=2x. So, 2x-40=180 3. 180+40=220. So. 2x=220 and X=110 4. Since x=110, place this number on (x+20) & (x-60). 110+20=130. 110-60=50. ANS: 50 and 130.

When Dahlia's professor eliminated the lowest score of her 4 quiz scores, her quiz average rose from 77 to 91. What was the score of the quiz that the professor eliminated?

Get the average score and multiply it by the total sum of quiz. 1. 77x4=308. 91x3=273. 2.308-273=35.

The sum of the interior angles of a regular polygon is less than 540°. Which could be a polygon? Indicate all such polygons. a. triangle b. quadrilateral c. pentagon d. hexagon

In a regular polygon with n sides, the formula for the sum of the interior angles is (n-2) x 180. This equation can be set up as an inequality to solve for n: (n-2) x 180 <540°. Dividing both sides by 180 gives us: n-2<3 2. n<5. 3. The polygons listed with fewer than 5 sides are the triangle and quadrilateral.

The average (arithmetic mean) of a, b, and c is 70 and the average (arithmetic mean) of d and e is 120. What is the average (arithmetic mean) of a, b, c, d, and e?

The average formula is: Average=Sum of terms/# of terms Since the average of a+b+c=70 we multiply x 3 (3 terms) and this gives us 210. d and 3 the average=120 so if we multiply x 2, this give us 240. Now get 210 + 240 and divide by 5 and your ANS: 90.

The average of all the consecutive integers from a to b inclusive is 39. Which of the following could be a & b? Indicate all such integers. a. 4 & 74 b. 19 & 39 c. 25 & 53 d. 29 & 59 e 33 & 45

The average of a group of consecutive integers is = to the average of the smallest & largest integers, so any pair of #'s whose average is 39 could be a & b. Since the average of a & b is equal to their sum divided by 2, any pair of numbers whose sum 2x39=78 is a valid answer choice. ANS=A, C,E.

Telephone Co. A charges $3.00 for the first minute of any long distance call and $0.50 for each additional minute. Telephone Co. B charges $2.00 for the first minute of any long distance call and $0.70 for each additional minute. If the cost of a call lasting x minutes, where x is a positive integer, is $15.00 more with Telephone Co. B than Telephone Co. A, then what is the value of x?

The cost of a telephone call lasting x minutes with TA is $3.00 for 1 minute and $0.50 for each additional minute of the additional x-1 minutes. So, the cost of this telephone call with TA is 3+0.5(x-1) dollars. The cost for TBis 2 +0.7(x-1) dollars. Since the cost of a call lasting x minutes with TB is $15 more than cost of a call lasting x minutes with TA we have this equation: 2 + 0.7(x-1)=3+0.5(x-1)+15. Therefore: 2+0.7x-0.7=3+0.5x-0.5=15 1.3+0.7x=0.5x+17.5 0.2x=16.2 x=81

The ratio of ½ to ³/₅ is the same as which of the following ratios? a. 1:5 b. 3:10 c. 2:3 d. 5:6 e. 3:2

The expression is the same as the value of 1/2 divided by 3/5. To divide a fraction, you multiply by the reciprocal, 1/2 x 5/3=5/6.

A square is inscribed inside a shaded circle. The circumference of the circle is 6π√2. What is the area of the shaded region?

The problem asks for the area of the shaded region so find the difference between the area of the circle and the area of the square. 1. Area of circle: find the radius. The circumference of the circle is 2πr. Since the circumference is given as 6π√2: 2. 2πr=6π√2 then, 2r=6√2 and then r=3√2. 3. The area of the circle is πr²=π(3√2)²=18π

The length of one side of a triangle is 12. The length of another side is 18. Which of the following could be the perimeter of the triangle. Indicate all such perimeters. a. 30 b. 36 c. 44 d. 48 e. 60

The problem asks for the perimeter of the triangle. The triangle inequality theorem states that the length of the third side of a triangle must be between the positive difference and the sum of the other two sides. So once you find the range of possible lengths for the 3rd side, you can add the side lengths to find the range possible. The 3rd side must be > than the difference 18-12=6 & less than the sum 12+18=30. So, the perimeter must be > than 12+18+6=36. and < than 12+18+30=60. ANS=C & D

A flower shop sells flowers in a ratio of roses to carnations of 5:2. The ratio of carnations to tulips sold is 5:3. What is the ratio of roses to tulips?

The question asks for a combined ratio. To compare ratio's the # for carnations must be the same in each ratio. Since carnations are represented by 2 in one ratio and 5 in the other, convert the ratios so that both have carnations represented by 10. The number 10 is chosen because it is the least common multiple of 2 and 5. The ratio of roses is 5:2=25:10 & the ratio of carnations to tulips is 5:3=25:6. Ans.=25:6

If 3x+y=-1, and y-2x=4, what is the value of x+2y?

The question asks for x +2y, but not for x or y individually. In these situations, it is not necessary to solve for x or y. 1. Rearrange the equations so that like terms are on top of each other and then add.

If x<-3 & x² + 5x+12=8, what is the value of x+2?

This is a quadratic equation, thus all terms are to be moved to one side and set equal to zero to solve for x. 1. x²+5x+(12-8)=0->x²+5x+4=0. 2. x²+5x+4=0 can be factored to: (x+4)(x+1)=0 3. Solving for X gives x=-1 or -4. The problem states that x<-3 so -4 is your value for x. 4. x+2=-4+2=-2. Answer is -2.

Point A (4,6) lies on a line with a slope of ⁻³/₄. Point B lies on the same line & is 5 units from Point A. Which of the following could be the coordinates of Point B?

Use the definition of slope as m=rise/run: 1. Point A vertically moves to 3 & horizontally to -4. Thus: (4-4, 6+3)=0,9 which will also lie on the line with slope -³/₄. 2. You can also move vertically to -3 and horizontally to positive 4 thus: (4+4, 6-3)=8,3). ANS: (0,9) & (8,3).

What is the probability of rolling a number greater than 2 twice in a row on a fair six-sided die, with each of the numbers 1-6 on each side?

Use the probability formula: 1. P=# of desired outcomes/# of total possible outcomes. 2. # of possible outcomes=6. 3. Probability of rolling a number > 2=4. 4/6=2/3. Probability of rolling a number >2 on the second roll is 4/6=2/3. 4. 2/3 x 2/3=4/9. Answer is 4/9.

What is the sixth term in a sequence in which the nth term is n(n-1)²?

Use the rule given, substituting 6 for n to find the sixth term. Be sure to follow the order of PEMDAS: 6(6-1)²=6(5)²=6(25)=150

A principle as 4 different trophies available for display in his two display cases. If only one trophy can fit in each display case, how many distinct ways are there to display two of the trophies in the cases at any given time?

this is a permutation question. Call the trophies A,B,C,D. You can arrange each in 3 different ways. 3 x 4=12. Answer=12.