Foundations of Math Flashcards
If , then x =
- -1.4
- -0.8
- 0.2
- 0.8
- 1.4
If we multiply both sides of the equation by (x + 2), we get 1.5x + 3 = 1.8.
If we multiply both sides of the equation by 2, we get 3x + 6 = 3.6
Further simplifying, 3x = -2.4, so x = -0.8.
The correct answer is B (2).
what is one possible value of x?
- -4
- -3
- -2
- 2
- 3
We can put this quadratic into standard form (ax2 + bx + c = 0) by first multiplying both sides of the equation by 2 and then subtracting 6.
x2 – 2x – 2 = 6
x2 – 2x – 8 = 0
We can now factor this quadratic as (x – 4)(x + 2) = 0
This means that x = 4 or -2.
The correct answer is C.
If and , then
- -4
- -2
- 0
- 2
- 4
One way to solve this problem is to use substitution. Solve the first equation for y, then substitute the expression into the second equation.
3x + y = 6
y = 6 – 3x
x + 3y = 10
x + 3(6 – 3x) = 10
x + 18 – 9x = 10
-8x = -8
x = 1
Now substitute 1 into either equation to solve for y:
3(1) + y = 6
y = 3
Finally, find the value of the expression you are asked for:
x – y = 1 – 3 = -2
The correct answer is B (2).
If -2 < x < 3, which of the following CANNOT be true?
- 3x < -3
- 2x + 2 < 4
- 3x < 8
- 3x > -3
- -3x > 8
We must check each of the ranges in the answer choices to see which one does not contain values within the given range -2 < x < 3.
(A) 3x < -3
x < -1
Has values within the range -2 < x < 3.
(B) 2x + 2 < 4
2x < 2
x < 1
Has values within the range -2 < x < 3.
(C) 3x < 8
x <
Has values within the range -2 < x < 3.
(D) 3x > -3
x > -1
Has values within the range -2 < x < 3.
(E) -3x > 8
x < (must change direction of inequality symbol when dividing by a negative)
Does NOT have values within the range -2 < x < 3.
The correct answer is E.
- 0.205
- 0.232
- 0.25
- 0.3
- 0.7
Calculate the squared and the cubed term first:
2(0.1) + 3(0.1)2 + 2(0.1)3 = 2(0.1) + 3(0.01) + 2(0.001)
Multiply each term by its coefficient, then sum the terms:
2(0.1) + 3(0.01) + 2(0.001) = 0.2 + 0.03 + 0.002 = 0.232
The correct answer is B.
- -3/2
- -3/5
- 3/5
- 3/4
- 10/3
First, simplify the denominator of the big fraction. Perform the subtraction by finding a common denominator:
Now divide, by taking the reciprocal of the bottom fraction:
The correct answer is E.
Among the 1,600 students at Hamilton High School, 45% take science courses. If of the students taking science courses are taking physics, how many students at Hamilton High School are taking physics?
- 300
- 360
- 400
- 600
- 720
We can solve this problem in two steps by first finding the number of students taking science courses and second finding the number of those students taking physics.
The number of students taking science courses = 0.45(1,600) = 720.
Thus, the number of students taking physics = 720( ) = 300.
The correct answer is A.
All of the following are less than EXCEPT
- 9/11
- 13/18
- 3/4
- 5/7
- 17/22
There are a couple of ways to compare fractions:
I. Compare the decimal equivalents of the fractions.
is equal to 0.111… (since it is of ). then, is equal to 0.777…, 7 times .
II. Use the cross multiplication trick:
because 3 x 3 = 9, which is greater than 2 x 4 = 8.
(A) because 9 x 9 = 81, which is greater than 7 x 11 = 77.
(B) , which is greater than .
(C) and is therefore less than .
(D) because 7 x 7 = 49, which is greater than 5 x 9 = 45.
(E) because 7 x 22 = 154, which is greater than 9 x 17 = 153.
The correct answer is A.
In the figure below, what is the value of x?
- 90
- 100
- 110
- 120
- 130
We can label many of the other angles in the figure. When two lines intersect, the opposite angles are equal; the upper right angle of the triangle in the figure is therefore 60 degrees. The angles that form a straight line sum to 180 degrees; the bottom angle of the triangle is therefore 180 – 140 = 40.
Since the sum of the angles in a triangle equals 180 degrees:
- y* + 60 + 40 = 180
- y* + 100 = 180
- y* = 80
x = 180 – y = 180 – 80 = 100
The correct answer is B.
What is the area of the trapezoid pictured below?
- 22.5
- 24
- 27
- 45
- 54
Since the triangle is a right isosceles triangle, the other leg of the triangle (the height of the trapezoid) must be 3. The top base if the trapezoid must be 6 since it is the opposite side of a rectangle.
The area of a trapezoid = (base 1 + base 2) × (height)
The area can also be found here by breaking up the figure into a rectangle (area of 6 × 3 = 18) and a triangle (area of ½ × 3 × 3 = 4.5) and adding these two areas. The correct answer is A.
Area of Circle - Fractions
n the circular region shown, with center O, the two unshaded sections constitute and of the area of the circle. What is x?
- 150
- 160
- 170
- 190
- 210
If the two unshaded sections constitute and of the area of the circle, then the shaded part of the circle covers the remaining fraction of the total area. We can compute this fraction by adding the fractions and , then subtracting that sum from 1:
+ = + =
Thus, the shaded part of the circle covers 1 – = .
Since a circle covers 360°, the angle x is of that 360°. Therefore, x = ( )×360° = 190°.
The correct answer is D.
If |a| = and |b| = , which of the following CANNOT be the result of a + b?
- -1
- -1/3
- 1/3
- 2/3
- 1
Given that |a| = , the value of a could be either or . Likewise, b could be either or . Therefore, four possible solutions to a + b exist, as shown in the following table:
- a*
- b*
- a* + b
1
-1
is the only answer choice that does not represent a possible sum of a + b.
The correct answer is D.
Which of the following is equivalent to x4 + x3 for all values of x?
- x7
- 2x7
- x4(x + 1)
- x3(x + 1)
- x3(x2 + 1)
All of the answer choices are in a different form than the original expression, x4 + x3, so the two terms must be combined somehow. When you need to add two terms with the same base, take a common term out of each. In this case, x3 is common to both of the terms.
When you pull an x3 term out of x4, you are left with one x and when you pull an x3 term out of x3, you are left with the number 1:
- x*4 + x3
- x3(x* + 1)
The correct answer is (D).
If x is an even integer and y is an odd integer, then which of the following CANNOT be an even integer?
- x+2y
- 2(x+y)
- 2x+y
- x/y
- xy
One way to approach this problem is to plug an even number for x and an odd number for y, and eliminate any answers that come up even. Let’s have x = 2 and y = 3.
(A) x + 2y = 2 + 2(3) = 8. Eliminate.
(B) 2(x + y) = 2(2 + 3) = 10. Eliminate.
(C) 2x + y = 2(2) + 3 = 7. Keep.
(D) x/y = . Keep.
(E) xy = 2(3) = 6. Eliminate.
Now, to decide between C and D, we can try additional numbers, with the goal of making one or the other expression equal to an even number. If we try x = 6 and y = 3, then we get . This eliminates D.
Alternatively, you can look at the structure of the expressions. In C, the correct answer, we have 2x + y. 2x is even for all integers x. Then, any even number plus any odd number will always be an odd number.
The correct answer is C.
Each of the 80 writers in a certain club is either exclusively left-handed or exclusively right-handed. If there are 12 more left-handed writers in the club than right-handed writers, how many of the writers are not left-handed?
- 28
- 32
- 34
- 46
- 52
Begin by assigning variables to the unknown quantities:
L = left-handed writers R = right-handed writers
Then, write equations using the information given in the problem.
L + R = 80 and L = R + 12
Substitute the second equation into the first equation to solve for R as follows:
L + R = 80
(R + 12) + R = 80
2R = 68
R = 34
The question asks for the number of writers who are NOT left-handed which is the same as asking for the number of right-handed writers (34). The correct answer is C.
