Math Sat

If x and y are positive integers greater than 1, and wx/y+w=1, then w = A) x-1/y-1 B) y/x-1 C) x+1/y D) x-y/x+y E) y/x+1

Let's say w=8, x=20, and y=152 (following the rules). Therefore, w=8, so look for the answer that gives 8! Only (B) works.

Rule: Whenever you have a two-variable equation in which one side is much more complicated than the other,

MAKE UP THE VARIABLE ON THE MORE COMPLICATED SIDE.

Rule: when the number you make up IS in the answer choices,

make it as insanely easy as you possibly can.

The average (arithmetic mean) of six integers is 32. If the numbers are all different, and if none is less than 10, what is the greatest possible value of any of these integers? A. 127 B. 132 C. 137 D. 142 E. 147

sum of all #'s/6 = 32 sum of all #'s = 92 ** If you want to make one number in an average set as large as possible, all the other numbers must be as small as possible. ** We wan to make the other 5 numbers as tiny as possible. The smallest any number can be is 10. All the numbers need to be different, so the next will be 11, 12, so on. 10+11+12+13+14+x=192 60+x=192 x = 132 (B)

Rule: If there is a value in a problem that is NOT defined,

then you can make up that value.

If the sum of 5 consecutive odd integers is k, which of the following could be a value of k? I. 0 II. -1 III. -5 A. I only B. II only C. III only D. I and II only E. II and III only

0 = no odd integers added produce 0 Eliminate (A) and (E). -1 = Nothing works to produce -1 Eliminate (B) and (D). You are left with (C). -5+-3+-1+1+3 = -5

The 3 Ultimate Math Strategies

1). Come up with your own numbers whenever possible. 2). If you can't come up with your own numbers, insert variables into the question. 3). If you can't insert your own numbers or your own variables, steal from the answer choices.

Math Guessing Strategy

1). Find 3 or more answer choices that share a common property, and kill the answer choices that don't share the property. 2). Repeat step 1 multiple time until you're left with the answer that's (probably) right. **Note: Once you eliminate answers, you can still use them for your elimination process.

Three lines are drawn in a plane. Which of the following CANNOT be the total number of points of intersection? A. 0 B. 1 C. 2 D. 3 E. They all could

(E) All of the answer choices are possible.

On a map, 1 centimeter represents 6 kilometers. A square on the map with a perimeter of 16 centimeters represents a region with what area? a. 64km² b. 96km² c. 256km² d. 576km² e. 8216km²

A square with a perimeter of 16 has sides of 4cm. The question states that 1cm=6km, so each side of the square represents 24km. The area is 24², so 576km² (D).

Which of the following accurately defines all possible values of p-q if 15≤p≤30 and 7≤q≤19? A. -4≤(p-q)≤23 B. -4≤(p-q)≤30 C. 8≤(p-q)≤11 D. 15≤(p-q)≤23 E. 22≤(p-q)≤49

15-7≤(p-q)≤30-19 8≤(p-q)≤11 (C)

If -5<m<10 and 2<n<4, which of the following must be true for (m+n)? A. -3<(m+n)<14 B. -7<(m+n)<6 C. -5<(m+n)<12 D. 8<(m+n)<14 E. 12<(m+n)<14

-5+2<(m+n)<10+4 = -3<(m+n)<14 (A)

What is the slope of the line that passes through (0,0) and is perpendicular to the line that passes through (-2,2) and (3,3)? A. -5 B. -1/5 C. 0 D. 1/5 E. 5

3-2/3-(-2) = 1/5 take the negative reciprocal, which is -5 (E).

If (2x²+5x+3)(3x+1) = ax³+bx²+cx+d for all values of x, what is the value of c?

3x(2x²+5x+3)+1(2x²+5x+3) =6x³+15x²+9x+2x²+5x+3 =6x³+15x²+9x+2x²+5x+3 =6x³+17x²+14x+3 Since c represent the coefficient of the "x" term, x = 14.

5/√x-3 = 3 For x>3, which of the following equations is equivalent to the equation above?

5/√x-3=3 5=3√x-3 5/3=√x-3 25/9=x-3 25=9(x-3) (D)

If x+y=10 and x and y are distinct positive integers, what is the greatest possible value of xy? A. 25 B. 24 C. 21 D. 16 E. 9

A = 5+5 = NOT DISTINCT B = 6+4 = this works! STOP RIGHT THERE. asks for the greatest, and since B works, you move on. In problems that ask for the greatest or least values, start in the answer choices with the greatest or least values, and stop when the answer choice works.

How many lines can be drawn from one vertex of a cube to the other vertices of the cube, such that the lines are not parallel to any edge of the cube? a. 4 b. 6 c. 7 d. 24 e. 42

A cube has 6 faces and 8 vertices. From one vertex of the square, there are 7 potential lines that could be drawn, but some of them will be parallel to an edge, and therefore not count. Of the 7 potential lines, three are parallel to an edge, leaving 4 lines that first the requirements stated in the question (A).

If a is 4 greater than b and a²+b²=10, what is the value of ab? a. 13 b. 6 c. 3 d. -3 e. -6

A quick option is the try out the answers, and recognize that only (D) works with all of the requirements. Algebraically: From the question, we know that a=b+4. We can arrange that to be a-b=4 and square both sides and we get (a-b)²=4², which gives a²+b²-2ab=16, makes it obvious that we can replace a²+b² with 10, so 10-2ab=16, -2ab=6 ab=-3 (D)

If x<x³<x², then which of the following must be true? A. x<-1 B. -1<x<0 C. 0<x<1 D. X>1 E. x is not a real number

A). say x=-2 -2<-8<-4 = doesn't work B). say x=-1/2 -1/2<-1/8<1/4 = this works! (B) is the correct answer, you don't even need to move on.

If f(x) = 3x+8, for what value of a is f(a)=a? A. -4 B. -2 C. -3/8 D. 0 E. 3/8

A. does -4 = 3(-4)+8? -4=12+8 -4=-4 YES! (A) is correct.

If 3x+2y=11, and 2x+3y=17, what is the average (arithmetic mean) of x and y?

Add the two equations to get 5x+5y = 28. Then, diving each term by 5, we get x+y = 28/5. The average of x and y is x+y/2 = (28/5)/2 = 28/10 = 2.8 (B).

Let A, B, and C be three point on a place such that AB:BC = 3:5. Which of the following can be the ratio AB:AC? I. 1:2 II. 1:3 III. 3:8 A). I only B). II only C). III only D). I and III only E). I, II, and III

Assume AB=3 and BC=5. The least that AC can be is 2, if A is on line ↔BC, between B and C; and the most AC can be is 8, if A is on the line ↔BC, so that B is between A and C. If A is not on line ↔BC, AC can be any length between 2 and 8. Therefore, the ratio AB:AC can be any number between 3:2 (1.5) and 3:8. (I and III are true). It cannot be 1:3 (=0.33) (II is false). Statements I and III are true (D).

So far this year, Adam has played 30 games of chess and as won only 6 of the. What is the minimum number of additional games he must play, given that he is sure to lose at least one third of them, so that for the year he will have won more games than he lost? A. 25 B. 34 C. 57 D. 87 E. It is not possible for Adam to do this.

Assume Adam loses the minimum number, one-third of the remaining games. Say he loses x games and wins 2x games. Then, in total, he will have 6+2x wins and 24+x losses. Finally, 6+2x>24+x 2x>18 x>18, and so x is at least 19. He has at least 19 additional loses and 38 additional wins, for a total of 57 more games (C).

Randy bought four rare books that cost $130, $120, $80, and $75, respectively. If he paid 1/3 of the total cost immediately and the remainder in 3 equal payments, how much was each of the three equal payments? A. $45 B. $55 C. $70 D. $90 E. $135

BITE SIZE PIECES: Total Cost = 130+120+80+75=405 1/3 Cost = 405/3 = 135 Remainder = 405-135 = 270 3 equal payments = 270/3 = 90 (D)

In rectangle EFGH, line segment EF ⊥ line segment EG, and GH=4. Which of the following could be the length of EG? A. 1 B. 2 C. 3 D. 4 E. 5

Begin DRAWING EFGHa s described. If you draw EG, you'll see that it's the diagonal of the rectangle. The diagonal divides a rectangle into two triangles: AFG and AGH. Since the hypotenuse is the longest side of the triangle and line segment EG is the hypotenuse of both of these right triangles, line segment EG must be longer than 4 (E) --> 5.

A googol is the number that is written as 1 followed by 100 zeroes. If g represents a googol, how many digits are there in g²? A. 102 B. 103 C. 199 D. 201 E. 202

By definition, a googol is equal to 10¹⁰⁰. Therefore, g²=10¹⁰⁰*10¹⁰⁰ = 10²⁰⁰, which, when written out, is the digit 1 followed by 200 zeroes, creating an integer with 201 digits (D).

On a certain map that is drawn to scale, 1.5 centimeters is equivalent to 2 miles. If two cities are 35 miles apart, how many centimeters apart should they be on this map? A. 24.75 B. 26.00 C. 26.25 D. 45.00 E. 46.33

CREATE YOUR OWN VARIABLE C=centimeters M=miles 1.5C/2M = xC/35M C = 26.26 (C)

What is the circumference, in inches, of a circle with an area of 16π square inches? A. 2π B. 4π C. 8π D. 16π E. 32π

CREATE YOUR OWN VARIABLE C=πd or 2πr, so all we need to ind C, or circumference, is r (radius), then we multiply r by 2. A = πr² 16π=πr² √16=r r=4 c = 2πr = 2π(4) = 8π (C)

Tom's weight is 20 pounds less than twice Carl's weight. If together Tom and Carl weigh 340 pounds, how much does Tom weight? A. 120 B. 160 C. 180 D. 200 E. 220

CREATE YOUR OWN VARIABLE Tom's weight = T Carl's weight = C T+C = 340 C = 340-T T = 2C-30 T=2(340-T)-20 T=660-2T 3T=600 T=220 (E)

If 4.5 zots are equivalent to 1 zat, how many zats are equivalent to 36 zots? A. 8 B. 9 C. 12 D. 16 E. 81

CREATE YOUR OWN VARIABLE w=zots y=zats 4.5w/1x=36w/xw x= 8 zats (A)

Beth had planned to run an average of 6 miles per hour in a race. She had a very good race and actually ran at an average speed of 7 miles per hour, finishing 10 minutes sooner than she would have if she averaged 6 miles per hour. How long as the race?

CREATE YOUR OWN VARIABLE M=miles H=hours Original: M/H = 6 M/6 =6H/6 H = M/6 New: H-1/6 so M/(H-1/6) = 7 M = 7H-(7/6) M+7/6 = 7H H = M/7 + 1/6 M/6 = M/7 + 1/6 M = 6M/7 + 6/6 M/7 = 1 M=7 miles (B)

Both ∆ABC and ∆AED are equilateral. If AD=2/3 DC, then the perimeter of ∆AED is what fraction of the perimeter of ∆ABC? a. 1/3 b. 2/5 c. 3/5 d. 2/3 e. 4/5

Come up with some number for the sides of the triangles based on the ratios given in the question. If AD is 2, then DC is 3. The triangles are equilateral, so the perimeter of triangle AED is 6. The side of triangle ABC is 5, so the perimeter is 15. The perimeter of the triangle is 6/15, or 2/5 that of triangle ABC (B).

Selena has two types of records, worth a total of $41.75. Of her 25 records, b records are worth $1.25 each and the remaining d records are worth $2.30 each. Selena has no other types of records. Which of the following sets of equations can be solved to determine how many of each record type Selena has? (A) b+d=41.75 3.5(b+d)=25 (B) 3.55(b+d)=41.75 1.25(b+d)=2.30 (C) 3.55(b+d)=41.75 1.25b+2.30d=25 (D) b+d=25 1.25(b+d)=2.30 (E) b+d=25 1.25b+2.30d=41.75

Do not solve the equations sets, as the question does not seek the actual values of b and d. Deal with the number of albums first. Because band b represents the number of each type of album and because there are a total of 25 albums, b+d=25. Eliminate (A), (B), and (C). Now address the value. The B album costs $1.25 each and the d album costs $2.30 each; the grand total is $41.75. Answer choice (E) reflects the question.

What is the length of the diagonal that has vertices located at (a,b), (9,2), (9,8), and (1,8) in the xy coordinate system?

Draw the figure and fill in the information given in the problem. The pairs (1,8) and (9.8) form the top side of a rectangle with length 8. Point (9,8) and (9,2) form the right side of the rectangle with a length of 6. To find the length of the diagonal, we use the information we have to use the Pythagorean Theorem or recognize the 6-8-10 Pythagorean triple to see that the length of the diagonal is 8.

Points A, B, and C are three vertices of a triangle. The location of point X is such that AX=BX. Which of the following could be true? I. X is on side AB. II. X is inside ∆ABC III. X is outside ∆ABC A. I only B. III only C. I and II only D. II and III only E. I, II, and III

Draw the triangle, and then check each statement. Could X be on side AB? Sure, if it is the midpoint. Eliminate (B) and (D). Check statement II- could x be inside ∆ABC? Yes, the only restriction is that it is equidistant (at equal distances) from A and B. Eliminate (A). Although statement III seems "opposite" of statement II, due to the inside and outside locations, a point outside the of the triangle could also be equidistant from A and B. The correct answer is (E).

In ∆ABC, side AB=BC and side AB ⊥ to side BC. If BC=10, what is the area of the triangle? a. 10√2 b. 25 c. 50 d. 50√2 e. 100

Draw the triangle. From the information in the question, we know we have right triangle ABC, with equal sides of 10. Since it is a right triangle, the two legs are the base and the height. Use the area formula for a triangle: (.5)(10)(10)=50 (C).

Zenia drew a route on a map, starting with a 32 centimer line from her home due North to Anne's home. She continued the route with a 44-centimeter line due south to Ben's home, a 33 centimer line due West to Caleb's home, and a 28 centimeter line due East to Damon's home. What is the distance on the map, in centimeters, from Damon's home to Zenia's home?

Draw this one out. Because Anne's, Zenia's, and Beth's all lie in that order on a vertical line with a length of 44, and we know the length between Anne's and Zenia's is 32, the length between Zenia's and Beth's is 12. Moving due west from Beth's home creates a right triangle. Caleb's, Damon's, and Beth's all line on the horizontal line in that order. The length between Caleb's and Beth's is 33, so subtract the length between Caleb's and Damon's to see that the length between Damon's and Beth's is 5. Now we have a right triangle with legs of 5 and 12. Use the Pythagorean Theorem or remember the 5-12-13 triangle triplet to find hat the diagonal distance between Damon's and Zenia's home is 13.

If y<0 and (x+2)(y-4)=0, then x= a. -4 b. -2 c. 0 d. 2 e. 4

Either (x+2) or (y-4) must equal 0. Because y<0, you know that (y-4) cannot equal 0, Therefore, (x+2) must equal 0, and x must be -2 (B).

If the variable a, b, c, d, and e have distinct values and are listed in order form from least to greatest, which of the following could be true? I. b is the arithmetic mean II. e is one of the modes III. d is the median a. I only b. II only c. I and II d. I and III e. I, II, and III

Evaluate the information in the passage and check out the three statements. The question states that five variables have distinct values. The easiest to check are statements II and III. Statement II is false because no two numbers are the same. Eliminate (B) and (E). Statement III is false because c, the middle value, must be the median. This eliminated (C) and (D), leaving only (A) as the correct answer.

(a+b)²≤(a-b)²+36 In the equation above, 0≤a≤b. What is the greatest possible value of a?

Expand the left and right sides of the equation to get a²-2ab+b²≤a²-2ab+b²+36. Subtract a²-2ab+b² from both sides to get 4ab≤36. Divide both sides by 4 to get ab≤9. To make a as big as possible, b must be as small as possible, but still equal to or greater than a. If a and b are equal, then you can take the square root of both sides to see that a≤3, making that the biggest value of a is 3.

If the measures of the angles of a triangle are the ratio of 1:2:3, what is the ratio of the lengths of the sides? A. 1:2:3 B. 1:1:√2 C. 1:√3:2 D. 3:4:5 E. cannot be determined

For some number x, the measure of the angles are x, 2x, and 3x; so 180 = x+2x+3x 6x = 180 x = 30 Therefore, the triangle is a 30-60-90 triangle, and the ratio of the sides is 1:√3:2 (C).

The degree angle of the three angles of a triangle is an integer. Which of the following CANNOT be a ratio of the measures? A. 2:3 B. 3:4:5 C. 4:5:6 D. 5:6:7 E. 6:7:8

If the ratio were a:b:c, then 180 = ax+bx+cx = (a+b+c)x Since in each of the answer choices, the ratio is written in lowest terms, a+b+c must be a factor of 180. This is the case in choices A through D. Only choice (E), 6:7:8, fails, which if not a divisor of 180.

If m and n are integers, and m=n-2/n-2/n², then which of the following could be the value of m? I. -5 II. -3 III. -1

In this problem, I can make up the value of n to solve it much more quickly. Since the value of m depends on the value of n, and since the side with n is more complicated than the side with m, I want to focus on the complicated side first. say n=1 m = 1-2-2 = -5 = I works! next, say n=-1 m = -1+2-2 = -3 = II works! Does III work? If I try moving to the next integer, -2, I realize that the last part of the equation (2/n²) would be a fraction. If I try 3, then both of the (2/n) portions give non-integers. 4? 5? Worse and worse! I quickly realize that when making up possible values for n, anything other than -1 or 1 will not give me an integer value for m. Thus, the answer is (B), I and II only.

The length of a line segment with endpoints L and P is a positive integer less than 24. N is the midpoint of LP, M if the midpoint of LN, and O is the midpoint of NP. Which of the following could be the distance between M and P? a. 18 b. 17 c. 16 d. 15 e. 14

It is best to draw the line and plot the points according to what is described. If we do this carefully we should see that MP=3/4 LP, and remember the length of LP must be an integer less than 24. If we use the answer choices from the values of MP, we get the following length for LP: a. 24 b. 22(2/3) c. 21(1/3) d. 20 e. 18(2/3). Only (D) is less than 24.

If me is an integer and m, m+1, and m+2 are the lengths of the sides of a triangle, which of the following could be the value of m? I. 1 II. 10 III. 100 A. I only B. II only C. III only D. II and III only E. 1, II, and III

Just check each choice: is there a triangle whose sides are 1,2,3? No, the sum of any two sides of a triangle must be greater than the third side. Are there triangle whose sides are 10,11,12 and 100+101+102? Yes! II and III are true (D).

Cathy's average rate during the Boston marathon was 10 minutes a mile for the first b hours where b<4. In terms of b, how many more miles does Cathy have to run to complete the 26 mile race? (A) 26-6B (B)26-600B (C)6B-26 (D)26-(6/B) (D)(26-B)/6

Make up a number for b. Suppose b=2, which means that Cathy runs for 2 hours to 120 minutes, and then set up as a proportion to find how many miles she travels. So 10min/1mile=120min/xmiles, so x=12 miles. Now 26-12=14 miles left. Replace b with 2 in the answer choices, and the one that gives 14 is the correct answer (A).

If 4/5p is two times the value of q/10, where p and q are both positive integers, then p is what percent of q? a. 25% b. 400% c. 80% d. 25% e. 4%

Make up numbers. If you plug in 10 for p, then you can translate half of the question into the equation (4/5)(10)=2(q/10), which means q=40 Translate the phrase "p is what percent of q" into the equation p=(x/100)*q. Fill in the values you obtained to get 10=(x/100)*4, solve for x, giving x=25 (D).

Maria has 74 pebbles that she wants to divide into 25 piles. If the tenth pile is to have more pebbles than any other pile, what is the least number of pebbles Maria can put into the tenth pile?

One pile of 4 pebbles will leave 70 pebbles to divide among the remaining 24 pies, which we can do with a combination of 1, 2, and/or 3 pebbles. 3 couldn't be the biggest pile because we would need much more than one pile of 3 to get to 74 in only 25 piles. The question hinges upon the fact that we can't have fractional pebbles, and that we're only allowed one pile of the highest number.

What is the least of 3 consecutive even integers is the sum of these integers is 66? A, 20 B. 21 C. 22 D. 23 E. 24

Only A works 20+22+24=66 so the least has to be 20

If Anthony had 3 times as many marbles as he actually has, he would have 1/3 as many marbles as Billy has. What is the ratio of the number of marbles Anthony has to the number of marbles Billy has? A. 1:9 B. 1:3 C. 1:1 D. 3:1 E. 9:1

PLUGGING IN Say Anthony has 1 marbles. If he had 3 times as many, he'd have 3, and if 3 is 1/3 the number Billy has, he has 9. So the ratio of Anthony to Billy is 1:9 (A).

If a is a multiple of 3 and b is an odd integer, which of the following MUST be an odd integer? A. a/b B. ab C. a+b D. 2a+b E. a+2b

PLUGGING IN say a = either 3 or 6 (giving it both even and odd qualities) b = 3 A). 6/3 = 2 = NOT ODD B) 3*6 = 18 = NOT ODD C). a+b = 3+3 = NOT ODD D). 2(3)+3 = 9 3(6)+3 = 21 = THIS WORKS E). 3+2(3) = 3+6 = 9 6+2(6)= 6+12 = 18 = NOT ODD Only (D) works

Lou can drive 50 miles in a hours. If he must drive b miles at the same rate, in terms of a and b, how many hours will the trip take? A. a/50b B. 50/ab C. 50ab D. ab-b/25 E. ab/50

PLUGGING IN say a=5 and b=7 Set up proportions: 50mi/5hrs=7mi/xhrs x=7/10 Plug in the numbers in the answer choices, and only (E) produces 7/10

The price of a dress is reduced by 1/5. If the new price is reduced by 1/4, the resulting price is what fractional part of the original price? A. 1/20 B. 2/5 C. 9/20 D. 11/20 E. 3/5

PLUGGING IN Even when the question doesn't have any variables, you can plug in. When you don't have a starting number? Plug in! Say price of dress = $20 Reduced: 20-20/5 = 20-4 = $16 New Price: 16-16/4 = 16-4 = $12 Fraction: new price/original price = 12/20 = 3/5 (E)

If the sum of three consecutive odd integers is k. Then, in terms of k, what is the greatest of the 3 integers? A. k-6/3 B. k-3/3 C. k/3 D. k+3/3 E. k+6/3

PLUGGING IN k = 1+3+5 = 9 Greatest of the three integers = 5 A. gives 1 B. gives 2 C. gives 3 D. gives 4 E. gives 5 Only (E) works

Claire is c years old and is 6 years younger than Alan. In terms of c, how many years old will Alan be in 3 years? A. c-6 B. c-3 C. c+3 D. c+6 E. c+9

PLUGGING IN say c=12 So A=18+3=21 Only (E) works 12+9=21

If 4⁺ = m, what does 4²⁺⁺² equal in terms of m? A. 4m² B. 16m² C. 16m³ D. m²+4 E. m²+16

PLUGGING IN say t=2 so 4²=16=m 4²⁺⁺² = 4⁴⁺² = 4⁶ = 4096 A). 4m² = 4(16)² = 1024 B). 16m² = 16(16)² = 4096 C). 16m³ = 16(16)³ = 65536 D). m²+4 = 16²+4 = 260 E). m²+16 = 16²+16 = 276 Only (B) works

The elements of set f are all the integers less than 20 that are the products of exactly two different prime numbers. Which of the following is set f? A. {2,3,5,7} B. {2,3,4,5,7,9} C. {6,10,14} D. {6,10,14,15} E. {4,6,9,10,14,15}

PRIME FACTORS: STARS WITH 2, NOT 1 (A) {(1*2),(1*3),(1*5),(1*7)} ---> 1 is not a prime number = X (B) {(1*2),(1*3),(1*4),(1*5),(1*7),(1*9)} ---> 1 is not a prime number = X (C) {(2*3),(2*5),(2*7)} ---> good, but is it ALL? Let's see. (D) {(2*3),(2*5)(2*7)(2*5)} ---> good, but again, is it ALL? (E) {(2*2)..... wait = not two DISTINCT prime numbers We're left with (D).

If 2a=3b, 1/3c=6b²+2, and b>0, what is c in terms of a?

Plug in answers for the variables. Since b is in both equations, let's say b=2. 2a=3b 6b²+2=1/3c 2a=3(2) 6(2)²+2-=1/3c 2a=6 26=1/3c a=3 c=78 Plug a (3) in the answer choices, and the only one that gives 78 is (A).

If a is a nonnegative integer and b is a positive multiple of 4, how many distinct values of ab are less than 10? a. none b. one c. two d. three e. five

Plug in values for a and b. Because ab must be less than 10, you should start with the lowest possible values for each. The smallest possible value for b is 0, and the smallest possible value for b is 0, so ab=0. Plug in 1&2 for a, which produces new values for ab of 4 and 8, respectively, both of which are less than 10. If we try a=3, however, ab is 12, which is more than 10. There are no more distinct values. So the answer is (D), three.

The measures of the three angles of a triangle are in the ratio of 5:5:10, and the length of the longest side is 10. From this information, which of the following can be determined? I. The area of the triangle II. The perimeter of the triangle III. The length of each of three altitudes

Reduce the ratio 5:5:10 to 1:1:5, so the angle measures are 45-45-90. If you know the length of any side of a triangle, you can find the other sides and hence the area (I), the perimeter (II), and the lengths of the altitude (III), two of which are the legs. Statements I, II, and III are true (E).

The ratio of r to s is 3 to 4. The ratio of s to t is 2 to 9. What is the value of r to t? A. 1 to 3 B. 1 to 6 C. 2 to 9 D. 3 to 10 E. 4 to 5

Rule: Oftentimes, you can't make up ALL of the numbers at once. Some of the numbers will depend on the other numbers, so only make up the minimum amount of numbers necessary to figure out the other ones. say r=3 and s=4 so 4/t=2/9 2t=36 t=18 so r/t = 3/18 = 1/6 (B).

If the different of two numbers is greater than the sum of the numbers, which of the following must be true? A. Neither number is negative. B. At least one of the numbers is negative. C. Exactly one of the numbers is negative. D. Both numbers are negative. E. None of these statements must be true.

STEAL ANSWER CHOICES + MAKE UP NUMBERS A). not true. 5+7 is larger than 5-7 B). could be true. 5-(-7) is greater than 5+(-7) C). could be true D). doesn't have to be true (proven in B) E). Test (B) and (C) first. (-5)+(-6) is smaller than (-5)-(-6), so it is not true that exactly one of them could be negative - both of them could be - so (B) must be true.

If 3a=b, 2b=c, 3c=d, and abcd≠0, what is the value of d/a?

Say a=3, b=9, c=18, and d=54. d/a = 54/3 = 18.

The cost of 4 oranges is d dollars. At this rate, what is the cost of 40 oranges? A. d/40 B. 40/d C. 10d D. 20d E. 40d

Say d=20 $20/4oranges = $x/40oranges x=$200 only (C) works.

Each of the k girls in a club agreed to raise an equal amount of money to give to charity to which the club had pledged a total of x dollars. If p more girls then later join the club and also agree to raise an equal share of the pledges amount, how much less would each of the original club members have to raise, in dollars, than she had originally agreed to raise? A. x/k B. x/k+p C. px/k+p D. x(k+p)/k E. px/k(k+p)

Say k=10, x=100, and p=5 Original: 10 girls raising 100 dollars = $10 each Then: 10+5 girls raising 100 dollars, 100/15= $6.66 each. So each girls pays $3.33 less. Plug in the values into the answer choices and only (E) works.

To get to a business meeting, Joanna drives m miles in h hours, and arrived 1/2 hour early. At what rate should she have driven to arrive exactly on time? A. m/2h B. (2m+h)/2h C. (2m-h)/2h D. 2m/(2h-1) E. 2m/(2h+1)

Say m = 20 and h = 5. 20/(5+1/2) = 20/5.5 = 3.636363 Plug in values in the answer choices, and only (E) works.

In a class of 330 students, there are 60 more girls than boys. How many girls are there in the class?

Set a variable g for girls, and translate the information in the problem into an equation: g+(g-60)=330 2g-60=330 2g=390 g=195

A circular grass field has a circumference of 120√π meters. If Eric can mow 400 square meters of grass per hour, how many hours will he take to mow the entire field? A. 4 B. 5 C. 6 D. 8 E. 9

Since C=2πr, 2πr = 120√π r=(120√π)/2π = (60√π)/π Area: A = πr² = π [(60√π)/π]² = 3600π²/π² = 3600 sq meters. 3600/400 = 9 hours to mow the entire field (E).

If a-b=10, and a²-b²=20, what is the value of b? A. -4 B. -2 C. 2 D. 40 E. 100

Since a²-b² = (a-b)(a+b), then 20 = a²-b² 20 = (a+b)(a-b) 20 = (a+b)(10) a+b = 2 +a-b =+10 2a=12 a=6 so 6-b =10 -b=4 b = -4 (A)

If a/b=4/7, and b/c=14/15, then what is the value of a/c? A. 4/15 B. 7/15 C. 8/15 D. 4/7 E. 7/8

Since b=14 in b/c, let's make b also 14 in a/b. So that means, a would be 8. a/c = 8/15 (C).

If y is inversely proportional to x and directly proportional to z, and x=4 and z=8 when y=10, what is the value of x+z when y=20? A. 6 B. 12 C. 16 D. 18 E. 24

Since y is inversely proportional to x, there is a constant k such that xy = k. Then k = (4)(10) = 40, and 40 = x(20), so x=2. Also, since y is directly proportional to z, there is a constant m such that y/z = m. Then m = 10/8 = 5/4 and 5/4 = 20/z, so 5z = 180 and z = 16 so x+z = 2+16 = 18 (D).

IN the xy plane, lines a&c intersect at the point with coordinated (n,7/2). If the equation of line a is y=1/2x+5, and the equation of line c is y=1/3x+b, what is the value of b?

Solve for n in the equation for line a by replacing x with n and with with 7/2; n=-3. Since the point of intersections is now (-3,7/2), we can use the two values in the second equation to find b by replacing x with -3 and y with 7/2 to give us 7/2=1/3(-3)+b solving for b=9/2.

If a+b-c=d+6, c-b=8, and 3a=2-d, what is the value of a? (A) 2 (B) 4 (C) 8 (D) 12 (E) 16

Stack the equations and add like this: a+b-c=d+6 -b+c +8 +3a -d+2 =4a=16 a=4 (B)

If the absolute value of 2-3y < 2, what is one possible value of y?

The absolute value complicates this problem in that you'll have to solve twice: once assuming that 2-3y is less that 2 and another for when -(2-3y) is less than 2. Start with the first one: 2-3y<2 -3y<0 y>0 Solve the second scenario: -(2-3y)<2 -2+3y<2 y<4/3 Therefore, y can be any number less than 4/3 and greater than 0.

If y<0 and z≠0, each of the following could be true EXCEPT a. y²z²<0 b. yz²<0 c. y²z<0 d. yz<0 e. y³z<0

The easiest way to solve this problem is to PLUG IN numbers for y and z. Let's say y=-2 and z=3. The left side of the inequality in A is (-2)²(3)²=(4)(9)=36, in B is (-2)(3)²=(-2)(9)=18, in C is (-2)²(3)=12, in D is (-2)(3)=-6, and E is (-2)³(3)=-24. Since (B), (D), and (E) are all true, you can eliminate them. Now let's try y=-2 and z=-3. A is (-2)²(-3)²=36, and the left side of C is (-2)²(-3)=(4)(-3)=-12. Since (C) is now true, you can eliminate it as well, and therefore, through process of elimination, (A) is the answer.

A fire-fighter with a 52 foot ladder approaches a burning building, which rests on level ground. If the top of the ladder must be at least 48 feet above ground level, what is the maximum possible distance, in feet, from the exact middle of the ladder to the wall of the building? a. 0 b. 5 c. 10 d. 15 e. 20

The latter makes a triangle wit the wall and the ground. Draw it out and label the information from the question. One hypotenuse (the leaning ladder) is given first at 52, and the height is 48. We need to know the width of the middle of the ladder, so we need the base of a triangle with half the height of half the hypotenuse. The smaller numbers will also make the Pythagorean theorem easier to work with: 24²+b²=26². You can go on to solve for b, or recognize that this is a 5-12-13 Pythagorean triplet, only twice as big. So the missing value is 5, which means b=10 (C).

What is the measure, in degrees, of the smaller angle formed by the hour hand and the minute hand of a clock at 11:20? A. 120 B. 130 C. 135 D. 140 E. 150

The minute hand, of course, is forming right at 11. The hour hand, however, is not. It was pointing at 11 20 minutes ago, or 1/3 hour ago. The hour hand if one-third of the way between 11 and 12, so there are 20 degrees between the hour hand and 12 and another 120 degrees to 4. 20+120=140 (D).

If a, b, and c are distinct positive odd integers less than 6, how many different values of a(b∧c) are possible? a. two b. three c. four d. five e. six

The positive odd integers less than 6 are 1, 3, and 5, so those are the possible values of a, b, and c. The question mentioned that a, b, and c, are distinct, which means they're each a different integer. Now try out all the possible arrangements: 1(3⁵)=243 1(5³)=125 3(1⁵)=3 3(5¹)=15 5(1³)=5 5(3¹)=15 The question asks for how many different values there are, so don't count a number twice. There are five different values (D).

Two cylindrical tanks have the same height, but the radius of one tank equals the diameter of the other. If the volume of the larger is k% more than the volume of the other, k= A. 50 B. 100 C. 200 D. 300 E. 400

The volume of the small tank is πr²h, and the volume of the larger tank is π(2)²h, which equals π4h, so the large tank is 4 times the size of the smaller one. BE CAREFUL! This is an increase of 300%, not 400%. 4 is 3 more than 1, so k=300.

How many primes less than 1000 are divisible by 7? A. none B. 1 C. more than 1 but less than 42 D. 142 E. more than 142

There is only 1 prime divisible by 7, namely, 7 (B).

If x is a positive integer, which of the following is equivalent to (2k^1/2)⁻²? A. 2/k B. 1/2k C. 1/4k D. 1/2k² E. 4/k

This could be solved algebraically if you're comfortable with root and squaring algebra, but let's try something different. Any number to the (1/2) power is its own square root, so say k=25. (2*25^1/2) = 10⁻² = 1/100. A) = 2/25 B) = 1/50 C) = 1/100 D) = 1/1250 E) 4/25 Only (C) works.

Each student in a cooking class of 50 students is assigned to create a dessert, and appetizer, or both. The total number of students creating an appetizer is 7 more than the number of students creating a dessert. If the number of students who create two dishes is the same as the number of students who create exactly one dish, how many students created only a dessert? (A)9 (B)16 (C)25 (D)34 (E)41

This question is easily answered using the group formula: group1+group2-both+neither=total#. In this case, we don't need the "neither" portion. We know the number of appetizers is 7 more than the number of desserts, and that half of the entire class made both dishes. This gives us d+(d+7)-25=50. Solving the equation gives d=34, but that is the number of everyone who made a dessert, not those who only made a dessert. Subtract the "both" number from both number, 34-25= 9 (A).

Three times a number is the same as that number subtracted from 12. What is that number?

Three times a number is the same as 3x. That number subtracted from 12 is the same as 12-x. Now solve: 3x=12-x 4x=12 x=3

On the critical reading portion of the SAT, the raw score is calculated as follows: 1 point is awarded for each correct answer, and 1/4 point is deducted for each wrong answer. If Ellen answers all q questions of the test and earned a raw score of 10, how many questions did she answer correctly? A. q-10 B. q/5 C. (q/5)-10 D. (q-10)/5 E. 8+(q/5)

To earn 10 points, Ellen needed to get 10 correct answers and then earn no more points on the remaining q-10 questions. To earn no points on a set of questions, she had to miss 4 questions [thereby losing 4*(1/4) =1 point] for every 1 question she got right in that set. She answered 1/5 of the q-10 questions correctly :and 4/5 of the term incorrectly). The total # of correct answers was 10+(q-10)/5 = 8+(q/5) (E). Alternate Solution: Let c be the number of questions Ellen answered correctly, and q-c be the number she missed. Then her raw score is c-1/4(q-c) = 40 5c-q = 40 5c = 40+q so c = (40+q)/5 = 8+(q/5) (E).

One deck of cards consists of six cards numbered 1 through 6, and a second deck consts of six cards numbered 7 through 12. If one card is chosen at random from each deck, and the numbers on these cards are multiplied, what is the probability that this product is an even number?

To simplify the problem, notice that half the cards in each deck are odd and half are even. Therefore, you only need to consider 4 possible outcomes: (even)(even), (even)(odd), (odd)(even), and (odd)(odd). Each of these outcomes is equally likely and the first three produce and even product. Therefore, the probability that the product is even is 3/4.

If 3x-8<12+5x, then A. x>10 B. x<10 C. x>-10 D. x<-10 E. x>0

WHEN YOU MULTIPLY OR DIVIDE BY A NEGATIVE IN AN INEQUALITY, YOU MUST FLIP THE SIGN. 3x-8<12+5x 3x<20+5x -2x<20 x>-10

-7, -5, -3, -1, 0, 1, 3, 5, 7 How many distinct products can be obtained by multiplying the numbers above? a. 9 b. 17 c. 19 d. 21 e. 31

Watch your signs and be sure to only count distinct products. Be methodical: start at the left and multiply that number by each of the other numbers, then do the same thing for the next number. Don't write down any numbers that aren't distinct. -7: 35, 21, 0, -7, -21, -35, -49 -5: 15, 5, -15, -25 -3: 3, -3, 9 0: no distinct products 1: -1 3, 5, and 7 have no distinct products. The number of distinct products is 17 (B).

Shawn creates a mean by mixing the pastas, sauces, and toppings in his kitchen. Each meal he creates consists of one type of pasta, one type of sauce, and one type of topping. If Shawn can make exactly 30 different meals, which of the following could NOT be the number of sauces Shawn has? (A)1 (B)2 (C)3 (D)4 (E)5

We can determine how many different meals Shawn can make my multiplying the number of pastas, sauces, and toppings: the product is the number of meals Shawn can make. Since Shawn can make 30 different meals, the number of sauces available to Shawn must be a factor of 30. Choice (D) therefore is the answer, as Shawn cannot have 4 different sauces to make 30 different meals. 4 does not divide into 30.

Right triangle A has base b, height h, and area x. Rectangle b has length 2b, and width 2h. What is the area of rectangle B in terms of x? A)2x B)4X C)6X D)7X E)8X

We can find the answer by replacing b and h with real numbers. Say b=3, and h=4. Since the area of a triangle is (1/2)*b*h, the area of a triangle is (.5)(3)(4)=6. The area of the triangle is x, so x=6. now find the area of the rectangle where b=3 and h=4: each quantity is doubled, so the rectangle is 6 by 8 = 48. Plug in x (6) into the answer choices, and the only one that gives 48 is answer choice (B).

(3a²b³)³ = A). 9a⁵b⁶ B). 9a⁶b⁹ C). 27a⁵b⁶ D). 27a⁶b⁹ E). 27a⁸b²⁷

We know that 3³=27, so we can kill (A) and (B). We also than that (a²)³=a⁶, so that kills (C) and (E), and we're left with (D).

If the sum of a and b is c, what is the average (arithmetic mean) of a, b, and c? A. 2c B. 2c/3 C. a+b/c D. 2c/a+b E. c-(a+b)

When there are variables in the question, and in the answer choices, try making up your own numbers. Let a=4 and b=5. That means b, the sum of a and b, is 9. The question asks for the average of a, b, and c: a+b+c/3=18/3=6. Only (B) works.

The wholesale price of a car is w dollars. The retail price of a car is r percent greater than the wholesale price. During a special promotion, the retail price of the car is then discounted by s percent. Which of the following expressions represents the price, in dollars, of this car during the special promotion? A. w(r/100)(s/100) B. w(1+r/100)(1-s/100) C. (rs/100) D. w+wr/100-ws/100 E. w(wr/100-ws/100)

When you see variables in the answer choices, try MAKING UP YOUR OWN NUMBERS to make the math easier. Let's say w=$100, and that the retail prices is 20% greater than the wholesale price, so r=20. Lastly, let's say that the special promotion provides a 15% discount, so s=15. Now work through the problem with these numbers: the markup on a car would be 20%, or $20, so the retail price is $120. During the special promotion, the $120 is discounted by 15%. What is 15% of 120? 120*15/100=$18, so the special promotion is 120-18=$102. Plug the values for the variables we used above in the answer choices and choose the one that results in $102. Only (B) works.

If a-b=-4, what is the value of a²-2ab+b²? A. -32 B. -16 C. 0 D. 16 E. 32

While you can solve this algebraically, this can be easily solved using real numbers. Say a=0 and b=4 0-4 = -4 0²-2(0)(-4)+(4)² = 16 (D).

Aaron scored 84 points on a test made up of 10 eight-point short answer questions and 8 five-point multiple choice questions. If no partial credit was given and he missed 2-short answer questions, how many questions did Aaron miss on the whole test?

Work this question into bite-sized pieces. If Aaron missed 2 short answer questions, out of 10 total short-answer questions, he answer 8 short-answer questions correctly. This means Aaron scored 8*8=64 points on the short-answer questions. If he scored 84 points total, then Aaron scored 20 points on the multiple choice questions. Because each multiple choice question is worth 5 points, Aaron scored 20/5=4 multiple choice questions correctly. There are 18 questions total, so Aaron answered 18-12=6 questions incorrectly.

The Tyler Jackson Dance Company plans to perform a piece that requires two dancers. If there are 7 dancers in the company, how many different pairs of dancers could perform the piece?

You could write out the pairs: first match A with the other letters to get AB, AC, AD, AG, AF, and AG. A will repeat after 6 matches, so it's out. B forms 5 pairs (BC, BD, BE, BF, BG), C forms 4 pairs, D forms 3 pairs, E forms 2 pairs, and F forms 1 pair. Add them all up, 6+5+4+3+2+1=21.

Container A and Container B are right circular cylinders. Container A has a radius of 4 inches and a height of 5 inches while container B has a radius of 2 inches and a height of 10 inches. Sarah pours equal amounts into both containers and finds that the height of the oil in Container A is 2 inches. What is the height of the oil in container B? a. 8 inches b. 8.5 inches c. 9 inches d. 9.5 inches e. 10 inches

You'll the volume of the cylinder formula from the box at the beginning of the math section: v=πr²h, when v is the volume, r is the radius, and h is the height. For Container A, the radius is 4 inches and the height of the oil is 2 inches, so the volume of the oil is v=π*4²*2= 32π. For Container B, the volume of the oil is the same but the radius is 2 inches, so put those numbers into the cylinder equation and solve for the height: 32π=π*2²*h, so 32π=4π. Divide both sides by 4π to get 8 (A).

What is the value of [4^1/2 * 8^1/3 * 16^1/4 * 32^1/5]^1/2.? A. 2 B. 4 C. 8 D. 16 E. 64

[√4*³√8*⁴√16*⁵√32]^1/2 (2×2×2×2)^1/2 (16)^1/2 = √16 = 4 (B).

an altitude of a triangle

a line segment through a vertex and perpendicular to a line containing the base.

In problems that ask for the largest/smallest value...

always start with the largest/smallest value.

If a and b are the lengths of the legs of a right triangle whose hypotenuse is 10 and whose area is 20, what is the value of (a+b)²? A. 100 B. 120 C. 140 D. 180 E. 200

a²+b²=10² a²+b²=100 and since the area is 20, 1/2ab=20 ab=40 and 2ab=80 Expand: (a+b)² = a²+2ab+b² = (a²+b²)+2ab = 100 + 80 = 180 (D).