SAT- Math Questions With All Correct
Answers
How do you convert from part-to-part ratios to part-to-whole ratios? Correct answer-If the parts add
up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number
in the original ratio over the sum of the numbers. For example, if the ratio of males to females is 1 to 2,
then the males-to-people ratio is 1/ (1+2) = 1/3 and the females-to-people ratio is 2/ (1+2) = 2/3
In other words, 2/3 of all the people are female.
How do you solve a proportion? Correct answer-To solve a proportion, cross multiply.
For example:
x/5 = 3/4
4x = 3 x 5
x = 15/4 = 3.75
How do you solve a problem involving rates? Correct answer-To solve a rate problem, use the units to
keep things straight. For example, if asked: If snow is falling at the rate of one foot every four hours, how
many inches of snow will fall in seven hours?
Your setup would be:
1 foot/ 4 hours = x inches/ 7 hours
12 inches/ 4 hours = x inches/ 7 hours
4x = 12 x 7
x = 21
What is the average rate and how do you find it? Correct answer-Average rate is not simply the
average of the rates.
Average A per B = Total A/ Total B, so
Average Speed = Total distance/ Total time
For example, to find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just
average the two speeds. First, figure out the total distance and the total time. The total distance is 120 +
120 = 240 miles. The times are three hours for the first leg and two hours for the second leg, or five
hours total. The average speed, then, is 240/5 = 48 miles per hour.
What is the formula used to find average? Correct answer-To find the average of a set of numbers,
add them up and divide by how many there are.
Average = Sum of the terms / Number of the terms
How do you quickly find the average of a series of evenly spaced numbers? Correct answer-For evenly
spaced numbers, the average is the middle value. To find the average of evenly spaced numbers, just
average the smallest and the largest.
For example, the average of all the integers from 13 through 77 is the same as the average of 13 and 77.
,13 + 77/ 2 = 90/ 2 = 45
If you have an average of a series of numbers, how can you find their sum? Correct answer-Use the
formula:
Sum = Average x Number of terms
If the average of 10 numbers is 50, then they add up to 10 x 50, or 500.
How do you find the missing number in a series when you are given the average? Correct answer-To
find a missing number when you're given the average, use the sum.
Sum = Average x Number of Terms
For example, if the average of 4 numbers is 7, then the sum of those 4 numbers is 4 x 7, or 28. Suppose
that 3 of the numbers are 3, 5, and 8. These 3 numbers add up to 16 of that 28, which leaves 12 for the
fourth number.
How do you find the median of a set of numbers? Correct answer-The median of a set of numbers is
the value that falls in the middle of the set.
For example, if you have 5 test scores, and they are 88, 86, 57, 94, and 73, you must first list the scores
in increasing or decreasing order: 57, 73, 86, 88, 94. The median is the middle number, or 86. If there is
an even number of values in a set (6 test scores, for instance), simply take the average of the two middle
numbers.
How do you calculate the total number of possibilities for several events to occur? Correct answer-The
fundamental counting principle: If there are m ways one event can happen and n ways a second event
can happen, then there are m x n ways for the 2 events to happen. For example, with 5 shirts and 7 pairs
of pants to choose from, you can have 5 x 7 = 35 different outfits.
How do you find the mode of a set of numbers? Correct answer-The mode of a set of numbers is the
value that appears most often.
For example, if your test scores were 88, 57, 68, 85, 99, 93, 93, 84, and 81, the mode of the scores would
be 93 because it appear more often than any other score.
If there is a tie for the most common value in a set, the set has more than one mode.
How do you multiply expressions with exponents that have a common base? Correct answer-To
multiply exponential terms with the same base, add the exponents and keep the same base:
x^3 x x^4 = x^3+4 = x^7
How do you calculate the probability that an event will take place? Correct answer-Number of
Favorable Outcomes/ Total Possible Outcomes
^ Probability =
For example, if you have 12 shirts in a drawer and 9 of them are white, the probability of picking a white
shirt at random is 9/ 12 = 3/ 4. This probability can also be expressed as .75 or 75%.
How do you divide expressions with exponents that have a common base? Correct answer-To divide
exponential terms with the same base, subtract the exponents and keep the same base:
y^13 / y^8 = y^13-8 = y^5
How do you raise a power to a power? Correct answer-To raise a power to a power, multiply the
exponents:
,(x^3)^4 = x^3 x 4 = x^12
How do you simplify square roots? Correct answer-To simplify a square root, factor out the perfect
squares under the radical. Take the square root of the perfect square and put the result in front.
How do you add and subtract roots? Correct answer-Only like radicals can be added/subtracted from
one another. To combine radicals, just add or subtract their coefficients.
How do you multiply and divide roots? Correct answer-The product of more than one square root is
equal to the square root of their total product.
The quotient of more than one square root is equal to the square root of their total quotient.
How do you calculate negative exponents and fractional exponents? Correct answer-A negative
exponent indicates a reciprocal. To arrive at an equivalent expression, take the reciprocal of the base
and change the sign of the exponent.
A fractional exponent indicates a root.
How do you determine the absolute value of a number? Correct answer-The absolute value of a
number is the distance of the number from zero on the number line. Because absolute value is a
distance, it is always positive.
The absolute value of 7 is 7. Similarly the absolute value of -7 is 7. Every positive number is the absolute
value of two numbers: itself and its opposite.
How do you evaluate an algebraic expression? Correct answer-To evaluate an algebraic expression,
substitute the given values for the unknowns and calculate according to the correct order of operations,
or PEMDAS.
For example, to find the value of x^2 = 5x - 6 when x = -2, replace x with -2: (-2)^2 + 5 (-2) - 6 = 4 + (-10) -
6 = -12.
How do you add and subtract monomials? Correct answer-To add or subtract monomials, first make
sure that the terms to be combined have exactly the same variable(s). This makes them like terms. Then,
to combine like terms, keep the variable part unchanged while adding or subtracting the coefficients.
For example: 2a + 3a = (2 + 3)a = 5a
How do you add or subtract polynomials? Correct answer-To add or subtract polynomials, combine
like terms, which are terms with exactly the same variable(s).
For example:
(3x^2 + 5x - 7) - (x^2 + 12) =
(3x^2 - x^2) + 5x + (-7 - 12) =
2x^2 + 5x - 19
How do you multiply monomials? Correct answer-To multiply monomials, multiply the coefficients
and the variables separately. Be sure to add the exponents when multiplying the bases.
For example: 2a x 3a = (2 x 3) (a x a) = 6a^2
How do you multiply binomials? Correct answer-To multiply binomials, use FOIL..
For example, to multiply (y + 3) by (y +4), first multiply the First terms: y x y = y^2. Next the Outer terms:
y x 4 = 4y. Then the Inner terms: 3 x 4 = 12. Then, combine like terms: y^2 + 4y + 3y + 12 = y^2 + 7y + 12.
, What is the general procedure for multiplying polynomials? Correct answer-If you want to multiply
polynomials with more than two terms, make sure you multiply each term in the first polynomial by
each term in the second. In other words, use the distributive property and then combine like terms.
For example:
(x^2 + 3x + 4)(x + 5) =
x^2(x + 5) + 3x(x + 5) + 4(x + 5) =
x^3 + 5x^2 + 3x^2 + 15x + 4x + 20 = x^3 + 8x^2 + 19x + 20
After multiplying two polynomials together, the number of terms in your expression before simplifying
should equal the number of terms in one polynomial multiplied by the number of terms in the second.
In the example, you should have 3 x 2 = 6 terms in the product before you combine like terms.
How do you factor the differences of two squares? Correct answer-One of the test maker's favorite
expressions to factor is the difference of squares.
a^2 - b^2 = (a + b)(a - b)
For example, x^2 - 9 factors to (x + 3)(x - 3)
How do you simplify an algebraic fraction? Correct answer-Simplifying an algebraic fraction is a lot like
simplifying a numerical fraction. The idea is to cancel factors common to the numerator and
denominator. Thus, simplifying an algebraic fraction begins with factoring.
For example, to simplify x^2 - x - 12 / x^2 - 9, first factor the numerator and denominator: x^2 - x - 12 /
x^2 - 9 =
(x - 4)(x + 3) / (x - 3)(x +3)... Canceling the factor x + 3 from the numerator and denominator leaves you
with
(x - 4) / (x - 3)
How do you solve a linear equation? Correct answer-To solve a linear equation, do whatever is
necessary to both sides to isolate the variable. For example, to solve the equation 5x - 12 = -2x + 9, first
get all the x's on one side by adding 2x to both sides: 7x - 12 = 9. Then, add 12 to both sides: 7x = 12.
Then, divide both sides by 7; x = 3.
What does solving "in terms of" mean? Correct answer-To solve an equation for one variable in terms
of another means to isolate the one variable on one side of the equation, leaving an expression
containing the other variable on the other side of the equation.
For example, to solve the equation 3x - 10y = -5x + 6y for x in terms of y, isolate x:
3x - 10y = -5x + 6y
3x + 5x = 6y + 10y
8x = 16y
x = 2y
How do you translate English expressions into algebraic expressions? Correct answer-To translate
from English into algebra, systematically turn phrases into algebraic expressions. This will allow you to
transform an algebraic sentence into an equation. Be careful about order, especially when a problem
calls for subtraction. Take the following example:
"The charge for a phone call is a flat rate of 'r' cents for up to minutes and 's' cents for each minute
thereafter. What is the cost, in cents, of a phone call lasting exactly 't' minutes?"
Answers
How do you convert from part-to-part ratios to part-to-whole ratios? Correct answer-If the parts add
up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number
in the original ratio over the sum of the numbers. For example, if the ratio of males to females is 1 to 2,
then the males-to-people ratio is 1/ (1+2) = 1/3 and the females-to-people ratio is 2/ (1+2) = 2/3
In other words, 2/3 of all the people are female.
How do you solve a proportion? Correct answer-To solve a proportion, cross multiply.
For example:
x/5 = 3/4
4x = 3 x 5
x = 15/4 = 3.75
How do you solve a problem involving rates? Correct answer-To solve a rate problem, use the units to
keep things straight. For example, if asked: If snow is falling at the rate of one foot every four hours, how
many inches of snow will fall in seven hours?
Your setup would be:
1 foot/ 4 hours = x inches/ 7 hours
12 inches/ 4 hours = x inches/ 7 hours
4x = 12 x 7
x = 21
What is the average rate and how do you find it? Correct answer-Average rate is not simply the
average of the rates.
Average A per B = Total A/ Total B, so
Average Speed = Total distance/ Total time
For example, to find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just
average the two speeds. First, figure out the total distance and the total time. The total distance is 120 +
120 = 240 miles. The times are three hours for the first leg and two hours for the second leg, or five
hours total. The average speed, then, is 240/5 = 48 miles per hour.
What is the formula used to find average? Correct answer-To find the average of a set of numbers,
add them up and divide by how many there are.
Average = Sum of the terms / Number of the terms
How do you quickly find the average of a series of evenly spaced numbers? Correct answer-For evenly
spaced numbers, the average is the middle value. To find the average of evenly spaced numbers, just
average the smallest and the largest.
For example, the average of all the integers from 13 through 77 is the same as the average of 13 and 77.
,13 + 77/ 2 = 90/ 2 = 45
If you have an average of a series of numbers, how can you find their sum? Correct answer-Use the
formula:
Sum = Average x Number of terms
If the average of 10 numbers is 50, then they add up to 10 x 50, or 500.
How do you find the missing number in a series when you are given the average? Correct answer-To
find a missing number when you're given the average, use the sum.
Sum = Average x Number of Terms
For example, if the average of 4 numbers is 7, then the sum of those 4 numbers is 4 x 7, or 28. Suppose
that 3 of the numbers are 3, 5, and 8. These 3 numbers add up to 16 of that 28, which leaves 12 for the
fourth number.
How do you find the median of a set of numbers? Correct answer-The median of a set of numbers is
the value that falls in the middle of the set.
For example, if you have 5 test scores, and they are 88, 86, 57, 94, and 73, you must first list the scores
in increasing or decreasing order: 57, 73, 86, 88, 94. The median is the middle number, or 86. If there is
an even number of values in a set (6 test scores, for instance), simply take the average of the two middle
numbers.
How do you calculate the total number of possibilities for several events to occur? Correct answer-The
fundamental counting principle: If there are m ways one event can happen and n ways a second event
can happen, then there are m x n ways for the 2 events to happen. For example, with 5 shirts and 7 pairs
of pants to choose from, you can have 5 x 7 = 35 different outfits.
How do you find the mode of a set of numbers? Correct answer-The mode of a set of numbers is the
value that appears most often.
For example, if your test scores were 88, 57, 68, 85, 99, 93, 93, 84, and 81, the mode of the scores would
be 93 because it appear more often than any other score.
If there is a tie for the most common value in a set, the set has more than one mode.
How do you multiply expressions with exponents that have a common base? Correct answer-To
multiply exponential terms with the same base, add the exponents and keep the same base:
x^3 x x^4 = x^3+4 = x^7
How do you calculate the probability that an event will take place? Correct answer-Number of
Favorable Outcomes/ Total Possible Outcomes
^ Probability =
For example, if you have 12 shirts in a drawer and 9 of them are white, the probability of picking a white
shirt at random is 9/ 12 = 3/ 4. This probability can also be expressed as .75 or 75%.
How do you divide expressions with exponents that have a common base? Correct answer-To divide
exponential terms with the same base, subtract the exponents and keep the same base:
y^13 / y^8 = y^13-8 = y^5
How do you raise a power to a power? Correct answer-To raise a power to a power, multiply the
exponents:
,(x^3)^4 = x^3 x 4 = x^12
How do you simplify square roots? Correct answer-To simplify a square root, factor out the perfect
squares under the radical. Take the square root of the perfect square and put the result in front.
How do you add and subtract roots? Correct answer-Only like radicals can be added/subtracted from
one another. To combine radicals, just add or subtract their coefficients.
How do you multiply and divide roots? Correct answer-The product of more than one square root is
equal to the square root of their total product.
The quotient of more than one square root is equal to the square root of their total quotient.
How do you calculate negative exponents and fractional exponents? Correct answer-A negative
exponent indicates a reciprocal. To arrive at an equivalent expression, take the reciprocal of the base
and change the sign of the exponent.
A fractional exponent indicates a root.
How do you determine the absolute value of a number? Correct answer-The absolute value of a
number is the distance of the number from zero on the number line. Because absolute value is a
distance, it is always positive.
The absolute value of 7 is 7. Similarly the absolute value of -7 is 7. Every positive number is the absolute
value of two numbers: itself and its opposite.
How do you evaluate an algebraic expression? Correct answer-To evaluate an algebraic expression,
substitute the given values for the unknowns and calculate according to the correct order of operations,
or PEMDAS.
For example, to find the value of x^2 = 5x - 6 when x = -2, replace x with -2: (-2)^2 + 5 (-2) - 6 = 4 + (-10) -
6 = -12.
How do you add and subtract monomials? Correct answer-To add or subtract monomials, first make
sure that the terms to be combined have exactly the same variable(s). This makes them like terms. Then,
to combine like terms, keep the variable part unchanged while adding or subtracting the coefficients.
For example: 2a + 3a = (2 + 3)a = 5a
How do you add or subtract polynomials? Correct answer-To add or subtract polynomials, combine
like terms, which are terms with exactly the same variable(s).
For example:
(3x^2 + 5x - 7) - (x^2 + 12) =
(3x^2 - x^2) + 5x + (-7 - 12) =
2x^2 + 5x - 19
How do you multiply monomials? Correct answer-To multiply monomials, multiply the coefficients
and the variables separately. Be sure to add the exponents when multiplying the bases.
For example: 2a x 3a = (2 x 3) (a x a) = 6a^2
How do you multiply binomials? Correct answer-To multiply binomials, use FOIL..
For example, to multiply (y + 3) by (y +4), first multiply the First terms: y x y = y^2. Next the Outer terms:
y x 4 = 4y. Then the Inner terms: 3 x 4 = 12. Then, combine like terms: y^2 + 4y + 3y + 12 = y^2 + 7y + 12.
, What is the general procedure for multiplying polynomials? Correct answer-If you want to multiply
polynomials with more than two terms, make sure you multiply each term in the first polynomial by
each term in the second. In other words, use the distributive property and then combine like terms.
For example:
(x^2 + 3x + 4)(x + 5) =
x^2(x + 5) + 3x(x + 5) + 4(x + 5) =
x^3 + 5x^2 + 3x^2 + 15x + 4x + 20 = x^3 + 8x^2 + 19x + 20
After multiplying two polynomials together, the number of terms in your expression before simplifying
should equal the number of terms in one polynomial multiplied by the number of terms in the second.
In the example, you should have 3 x 2 = 6 terms in the product before you combine like terms.
How do you factor the differences of two squares? Correct answer-One of the test maker's favorite
expressions to factor is the difference of squares.
a^2 - b^2 = (a + b)(a - b)
For example, x^2 - 9 factors to (x + 3)(x - 3)
How do you simplify an algebraic fraction? Correct answer-Simplifying an algebraic fraction is a lot like
simplifying a numerical fraction. The idea is to cancel factors common to the numerator and
denominator. Thus, simplifying an algebraic fraction begins with factoring.
For example, to simplify x^2 - x - 12 / x^2 - 9, first factor the numerator and denominator: x^2 - x - 12 /
x^2 - 9 =
(x - 4)(x + 3) / (x - 3)(x +3)... Canceling the factor x + 3 from the numerator and denominator leaves you
with
(x - 4) / (x - 3)
How do you solve a linear equation? Correct answer-To solve a linear equation, do whatever is
necessary to both sides to isolate the variable. For example, to solve the equation 5x - 12 = -2x + 9, first
get all the x's on one side by adding 2x to both sides: 7x - 12 = 9. Then, add 12 to both sides: 7x = 12.
Then, divide both sides by 7; x = 3.
What does solving "in terms of" mean? Correct answer-To solve an equation for one variable in terms
of another means to isolate the one variable on one side of the equation, leaving an expression
containing the other variable on the other side of the equation.
For example, to solve the equation 3x - 10y = -5x + 6y for x in terms of y, isolate x:
3x - 10y = -5x + 6y
3x + 5x = 6y + 10y
8x = 16y
x = 2y
How do you translate English expressions into algebraic expressions? Correct answer-To translate
from English into algebra, systematically turn phrases into algebraic expressions. This will allow you to
transform an algebraic sentence into an equation. Be careful about order, especially when a problem
calls for subtraction. Take the following example:
"The charge for a phone call is a flat rate of 'r' cents for up to minutes and 's' cents for each minute
thereafter. What is the cost, in cents, of a phone call lasting exactly 't' minutes?"
