SAT Math

John had $150. He used 85% of it to pay his electric bill and 5% of it on a gift for his mother. How much did he have left?

$150x.85=$127.50 $150x.05=$7.50 $127.50+$7.50=$135 $150-$135=$15

If x^2=2, what is the value of (x+1/x)(x-1/x)?

(x+1/x)(x-1/x) = x^2-1/x^2=2-1/2=1.5

What is the value of -2*3?

-2*3=(-2)(3)-(-2+3)=-6-1=-7

Jim receives a commission of 25 cents for every $20 worth of merchandise he sells. What percent is his commission?

1% of $20 is 20 cents and the percent is a little more than 1%

For any real numbers c and d, c*d is defined to be c^d+d^c. What is the value of 1*(2*3)?

2*3=2^3+3^2=8+9=17 1*17=1^17+17^1=1+17=18

For any numbers a and b, a:)b is defined as a:)b= a+b/a-b. What is the value of 25:)15?

25+15=40 25-15=10 40/10=4

Which of the following is NOT equivalent to 3/5? A) 24/40 B) 60% C) 0.6 D) 3/7 x 7/5 E) 3/7 divided by 7/5

3/7 divided by 7/5=3/7 x 5/7 = 15/47,, which is not equivalent to 3/5

What is the value of 1+7/5 / 1-5/7?

35(1+7/5) / 35(1-5/7) = 35+49/35-25= 8.4

By how many degrees does the angle formed by the hour hand and the minute hand of a clock increase from 1:27 to 1:28?

360/12=30 degrees 30/60=0.5 degrees The minute hand moves through 30 degrees every 5 minutes and 6 degrees each minute. In the minute, the hands increases by 6-.5= 5.5 degrees

In the figure, what is the value of x? A polygon with two right angles, the x between the right angles, and a 35 degrees angles across a right angle on the bottom.

360= 90+90+35+x=215+x x=360-215=145

If 3x+5y=14 and x-y=6, what is the average of x and y?

3x+5y=14 + x-y=6 4x+4y=20 x+y=5 x+y/2=5/2=2.5

The chart depicts the number of electoral votes assigned to each of the 6 New England states. What is the average (arithmetic mean) number of electoral votes, to the nearest tenth, assigned to these states?

4+4+3+13+4+8=36 36/6=6

If 5(3x-7)=20, what is 3x-8?

5(3x-7)=20 15x-35=20 15x=55 x=55/15=11/3(3)=11, which is x. Subtract 3x-8: 11-8=3

For what value of x does x*5=x*10?

5x-(5+x)=10x-(10+x) 5x-5-x=10x-10-x 4x-5=9x-10 5x=5 x=1

A jar contains 10 red marbles and 30 green ones. How many red marbles must be added to the jar so that 60% of the marbles will be red?

60%=3/5, you have 10+x/40+x = 3/5 5(10+x)=3(40+x)= 50+5x= 120+3x 2x=70 x=35

In the figure, what is the sum of the measures of all of the marked angles?

7 angles = 720 angles

If 7x+3y=17 and 3x+7y=19, what is the average(arithmetic mean) of x and y?

7x+3y=17 + 3x+7y=19 10x+10y=36 x+y=3.6 x+y/2=3.6/2= 1.8

In the figure, what is the value of x-y? One triangle with two triangles inside. A right angle on top, the x on the angle of the left triangle(ACD) and a right angle across that on the other triangle(CDB) near the center and the y by the right angle on top on the right triangle.

90+ <B + x = 90+<B +y x=y x-y=0

In the grid, what is the area of quadrilateral ABCD?

Area of trapezoid: A=1/2(b1+b2)h b1=CD=9, b2=AB=4, and h=3 1/2(9+4)(3)=19.5

Nadia will be X years from now. How old was she Z years ago? A) x+y+z B) x+y-z C) x-y-z D) y-x-z E) z-y-x

Assume she will be 10 in 2 years. How old was she 3 years ago? If she will be 10 in 2 years, she is 8 now and 3 years ago was 5. Which of the choices equals 5 when x=10, y=2, and z=3? C

From 2002 to 2003 the sales of a book decreased by 80%. If the sales in 2004 were the same as in 2002, by what percent did they increase from 2003 to 2004?

Assume that 100 copies were sold in 2002(and 2004). Sales dropped by 80(80% of 100) to 20 in 2003 and then increased by 80, from 20 back to 100, in 2004. Percent increase is actual increase/original amount x 100%= 80/20x100%=400%

In the figure, O is the center of the circle. If OA = 4 and BC = 2, what is the value of x?

BCD is a right triangle in which the hypotenuse is twice as long as one leg, which means this is a 30-60-90 triangle. The angle opposite that leg measures 30 degrees. x= 30 degrees

In the figure, Q is a point on the circle whose center is O and whose radius is r, and OPQR is a rectangle. What is the length of diagonal PR?

Draw a line across OQ. The two diagonals are congruent, and, since OQ is a radius, OQ=PR=r.

Tony drove 8 miles west, 6 miles north, 3 miles east, and 6 more miles north. How far was Tony from his starting place?

Draw diagram. 25+144=169 sq rt of 169 is 13 13

What is the area of quadrilateral ABCD?

Draw in diagonals AC, CD, and CE. You see that the area of triangle ACD is 1/2 (4) (4) =8, and the area of triangle BAC is 1/2 (6)(10) =30. Then the area of ABCD is 38

In the figure, the circle with center O is inscribed in square ABCD. Line segment AO intersects the circle at P. What is the length of AP?

Extend AO to form diagonal AC. Since ADC is an isosceles right triangle, AC=2 sq rt 2 and AO is half of that, or sq rt 2. Draw in diameter EF parallel to AD . Since diameter is 2, the radius is 1. AP=AO-PO=sq rt 2 -1

The wholesale price of potatoes is usually 3 pounds for $1.79. How much money, in cents, did a restaurant save when it was able to purchase 600 pounds of potatoes at 2 pounds for $1.15?

For 600 pounds the restaurant would have to buy 200 3-pound bags for 200x$1.79=$358. On sale, it bought 300 2-pound bags for 300x$1.15 =$345. The restaurant saved 13 dollars. 1300 cents

If the sum of five consecutive even integers is 740, what is the largest of those integers?

If 146 is the largest of the 5 integers, the integers are 146, 144, 142, 140, and 138. The sum is 710, which is 30 less than 740. Eliminate all choices that are one of the integers and chose the closest integer to be the largest.

In a rectangle AFJE, AE is 8 and the radius of the circle is 5. What is the area of the rectangle BGHC? CD and its correspondent is longer than AB and DE is longer than CD.

If AE is 8, AD is 4, AC is 2, and BC is 1. BG is longer than AD, which is 4, and shorter than AE, which is 8. Therefore, BG is more than 4 and is less than 8. 6

Anne drove for h hours at a constant rate of r miles per hour. How many miles did she go during the final 20 minutes of her drive? A) 20r B) hr/3 C) 3rh D) hr/20 E) r/3

If Anne drove at 60 miles per hour for 2 hours, how far did she go in the last 20 minutes? Since 20 minutes is 1/3 of an hour, she went 20(1/3 of an hour) miles. Only E=20 when r=60 and h=2.

The average of 5, 10, 15, and x is 20. What is x?

If the average of 4 numbers is 20, and 3 of them are less than 20, the other one must be greater than 20. Since 5 and 10 are a lot less than 20, x will be a lot more than 20.

In triangle ACB, what is the value of x?

If the hypotenuse of a right triangle is twice the length of one of the legs, the triangle is a 30-60-90 triangle, and the leg is 60 degrees.

A prize of $27,000 is to be divided in some ratio among three people. What is the largest share?

If the prize were divided equally, each share would be worth $9000. If it is divided unequally, the largest share is more than $9000.

If a=b+1/2, b=2c+1/2, and c=3d+1/2, which of the following is an expression for d in terms of a? A) a-2/6 B) 2a-3/6 C) 2a-3/12 D) 3a-2/18 E) 4a-3/24

Let d=1. Then c= 3 1/2, b=7 1/2, and a=8. Which of the choices equals 1 when a=8? Only A

In the figure, each side of square ABCD is divided into three equal parts. If a point is chosen at random inside the square, what is the possibility it will be in the shaded region?

Let x=1. Then the area of the whole square is 9. The total shaded area is 1 and the shaded area is 1/9

If P painters can paint H houses in 4 days, how many houses can 6 painters, working at the same rate, paint in 2 days? A) h/3p B) 3hp/4 C) hp/12 D) 3h/p E) 3p/h

Pick 2 numbers. Say p=1 and h=1. Then 1 painter can paint 1 house in 4 days. ^ painters can paint 6 houses in 4 days, and so in 2 days 6 painters can paint 3 houses. Which will be bigger if you replace 100 by h in D and E? D will be bigger!

If the sum of 4 consecutive odd integers is s, then, in terms of s, what is the greatest of these integers? A) s-12/4 B) s-6/4 C) s+6/4 D) s+12/4 E) s+16/4

Pick 4 consecutive odd integers like 1, 3, 5, and 7. The sum is 16. The greatest is 7, so plug in these choices and find which one =7. D

If a is equal to b multiplied by c, which of the following is equal to b divided by c? A) a/bc B) ab/c C) a/c D) a/c^2 E) a/bc^2

Pick numbers to satisfy a=bc like a=6, b=2, c=3. Then plug them in to each choice and find which one is equal to b/c

If a school cafeteria needs C cans of soup each week for each student, and if there are S students in the school, for how many weeks will X cans of soup last? A) cx/s B) xs/c C) s/cx D) x/cs E) csx

Replace C, S, and X with three numbers like 2,5, and 10. Since the cafeteria needs 2x5=10 cans of soup per week, 20 cans will last for 2 weeks. Find the choice where if u plug them in it equals 2. D

What is 3% of 4%?

Since 3% of a number is just a small part of it, 3% of 4% must be much less than 4%. 0.12%

At a speed of 48 miles per hour, how many minutes will be required to drive 32 miles?

Since 32/48=2/3, it will take 2/3 of an hour to drive 32 miles. 2/3(60)=40

If 25% of 220 equals 5.5% of w, what is w?

Since 5.5% of w equals 25% of 220, which is greater than 5.5% of 220, w must be greater than 220.

From 1990 to 2000, Michael's weight increased by 25%. If his weight was W kilograms in 2000, what was it in 1990?

Since his weight increased, his weight in 1990 was less than W.

The product of 3 positive integers is 300. If one of them is 5, what is the least possible value of the sum of the other 2?

Since one of the integers is 5, the product of the other 2 is 60(5x60=300). List all the pairs,(a,b), of positive integers whose product is 60, and check their sums. First, let a=1, then 2, and so on. 16

The average of 10 numbers is -10. If the sum of 6 of them is 100, what is the average of the other 4?

Since the average of all 10 numbers is negative, so is their sum. The sum of the first 6 is positive, so the sum of the others must be negative. -100

From 2003 to 2004 the number of boys in the school chess club decreased by 20%, and the number of girls in the club increased by 20%. The ratio of girls to boys in the club in 2004 was how many times the ratio of girls to boys in the club in 2003?

Since the problem involves percents, use 100. Assume that in 2003 there were 100 boys and 100 girls in the club. Since 20% of 100 is 20, in 2004 there were 120 girls(a 20% increase) and 80 boys(a 20% decrease). Make fractions of the ratios and reduce. E

In the figure, diagonal EG of square EFGH is one-half of diagonal AD of square ABCD. What is the ratio of the area of the shaded region to the area of ABCD? A) sq rt of 2:1 B)3:4 C)sq rt of 2:2 D)1:2 E)1:2 sq rt2

Since the shaded region is less than ABCD, the ratio must be less than 1. Since it is more than half of that square, the ratio is greater than 0.5. B

A jar contains only red and blue marbles. The ratio of the number of red marbles to the number of blue marbles i 5:3. What percent of the marbles are blue?

Since there are 5 red marbles for every 3 blue ones, there are fewer blue ones than red ones. Fewer than half of the marbles are blue.

In the figure ABC with <A with 50 degrees and line extending by <C of x, AB=AC, what is the value of x?

Since triangle ABC is isosceles, with AB=AC, the other two angles in the triangle, <B and <C, each measure 65 degrees. x+65=180 x=115

In the diagram, rays PA and PB are tangent to circle O. Which of the following is equal to z?

Tangents to a circle are perpendicular to the radii drawn to the points of contact, so w= y=90. The sum of the four angles in a quadrilateral is 360 degrees, so w+x+y+z= 360. Then 90+x+90+z=360. x+z=180. z=180-x

If f(x)=4x^2+2x^4, what is the value of f(-2)?

The answer is positive. 48

In how many ways can Al, Bob, Charlie, Dan, and Ed stand in line if Bob must be first and either Charlie or Dan must be last?

There are possibilities when C is last. There will be 6 more when D is last. There are 12 ways in all

If the average (arithmetic mean) of 2, 7, and x is 12, what is the value of x??

Try choices. Choice D is 27. 2+7+27/3=36/3=12

If a-b=1, b-c=2, and c-a=d, what is the value of d?

a-b=1 b-c=2 +c-a=d 0=3+d d=-3

A competition offers a total of $250,000 in prize money to be shared by the top 3 contestants. If the money is to be divided among them in the ratio of 1:3:6, what is the value of the largest prize?

x+3x+6x=$250,000 10x=$250,000 x=$25,000 6x=$150,000

How many positive numbers are solutions of the equation y*y=y?

y*y=y y^2-2y=y y^2-3y=0 y(y-3)-0 y=0 or y=3

If a-b+c=7 and a+b-c=11, which of the following must be true? I) a is positive II) b>c III)bc<0 A) none B)I only C) II only D) III only E) I and II only

a-b+c=7 +a+b-c=11 2a=18 a=9

The diagonal of square II is equal to the perimeter of square I . The area of square II is how many times the area of square I?

If the side of square I is 1, its perimeter is 4. Then the diagonal of square II is 4. Now use the formula A= 1/2d^2 to find the area of square II is 1/2(4)^2 =8

If 2 sq rt of 2x+1 and then + 5=8, then x =? A) -1/8 B) 0 C) 5/8 D) 1 E) 9/8

Plug in each choice to find 8 5/8

What is the largest integer, n, such that 112/2 sq rt of n is an integer? A) 1 B) 2 C) 3 D) 4 E) 5

Plug in each option of 9 to end up with an integer. 4

In triangle XYZ, if XY<YZ<ZX, then what is z?

Redraw the triangle so that it matches up to scale. Identify largest and smallest sides. z<60

In the square ABCD,it has been divided into four triangles by its diagonals. If the perimeter of each triangle is 1, what is the perimeter of the square?

Since triangle BEC is a 45-45-90 right triangle, BE and EC are each side sq rt 2/2. s sq rt 2 + s 1=s(sq rt 2 + 1) s= 1/ sq rt 2 + 1 P= 4s= 4/sq rt 2 +1

A region inside a semicircle of radius r is shaded. What is the area of the shaded region?

The area of a circle is 1/2pie^2 and the area of the semicircle is 1/2pier^2. The area of the shaded region must be less than 1/2pier^2.

In the figure, four semicircles are drawn, each centered at the midpoint of one of the sides of square ABCD. Each of the four shaded "petals" is the intersection of two of the semicircles. If AB =4, what is the total area of the shaded region? A) 8pie B)32-8pie C)16-8pie D)8pie-32 E) 8pie-16

The area of the square is 16 and the area of the shaded area is 9. Solve each choice and subtract it by 16 to equal 9.

In the figure, ABCD is a rectangle and BE and CF are arcs of circles centered at A and D. What is the area of the shaded region?

The entire region is a 2x5 rectangle whose area is 10. Since each white region is a quarter-circle of radius 2, the combined area of these regions is that of a semicircle of radius 2: 1/2 pier^2=2pie The area of the shaded region is 10-2pie=2(5-pie)

In a triathlon the athletes cover 1/24 of the total distance by swimming, 1/3 of it by running, and the rest by bike. What is the ratio of the distance covered by bike to the distance covered by running?

The least common denominator of the 2 fractions is 24. Assume that the total distance is 24 miles. The athletes swim for 1 mile and run for 8(1/3 of 24) miles. The remaining 15 miles they cover it by bike. 15:8

At Central High School each student studies exactly one foreign language. Three-fifths of the students take spanish, and one-fourth of the remaining students take italian. If all of the others take french, what percent of the students take french?

The least common factor of 3/5 and 1/4 is 20, so there are 20 students. Then the number of students taking spanish is 12(3/5 of 20). Of the remaining 8 students, 2(1/4 of 8) take italian. The other 6 take french. 6 is 30% of 20

A palindrome is a number, such as 935539, that reads the same forward and backward. How many palindromes are there between 100 and 1000?

Write down the numbers in the 100's that end in 1. Then write the numbers beginning and ending in 2. There are 10 numbers beginning with 1, and 10 beginning with 2, and there will be 10 beginning with 3,4...,9 for a total of 9x10=90 palindromes

If a and b are positive numbers, with a^3=3 and a^5=12b^2, what is the ratio of a to b?

a^5/a^3= 12b^2/3= 4b^2 and a^5/a^3= a^2. a^2 = 4b^2 a^2/b^2= 4 a/b=2