How to evaluate algebraic expressions - PSAT Math
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A rowing team paddles upstream at a rate of 10 miles every 2 hours and downstream at a rate of 27 miles every 3 hours. Assuming they are paddling at the same rate up and downstream, what is the speed of the water?
A rowing team paddles upstream at a rate of 10 miles every 2 hours and downstream at a rate of 27 miles every 3 hours. Assuming they are paddling at the same rate up and downstream, what is the speed of the water?
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Upstream: p – w = (10/2) or p – w = 5 miles/hour
Downstream: p + w = (27/3) or p + w = 9 miles/hour
Then we add the two equations together to cancel out the w's. After adding we see
2p = 14
p = 7 miles/hour where p is the rate of the paddling. We plug p into the equation to find
w = 2 miles/hour where w is the rate of the stream's water.
Upstream: p – w = (10/2) or p – w = 5 miles/hour
Downstream: p + w = (27/3) or p + w = 9 miles/hour
Then we add the two equations together to cancel out the w's. After adding we see
2p = 14
p = 7 miles/hour where p is the rate of the paddling. We plug p into the equation to find
w = 2 miles/hour where w is the rate of the stream's water.
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Tim is two years older than his twin sisters, Rachel and Claire. The sum of their ages is 65. How old is Tim?
Tim is two years older than his twin sisters, Rachel and Claire. The sum of their ages is 65. How old is Tim?
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The answer is 23.
Since Rachel and Claire are twins they are the same age. We will use the variable r to represent both Rachel and Claire's ages.
From the question we can form two equations. They are:
t = r + 2 and 65 = t + 2r
lets plug the first equation into the second to solve for r.
65 = (r + 2) + 2r
65 = 3r +2
63 = 3r
r = 21 This means Rachel and Claire are 21 years old. Plug this into the equation so
t = 23 Tim is 23 years old.
The answer is 23.
Since Rachel and Claire are twins they are the same age. We will use the variable r to represent both Rachel and Claire's ages.
From the question we can form two equations. They are:
t = r + 2 and 65 = t + 2r
lets plug the first equation into the second to solve for r.
65 = (r + 2) + 2r
65 = 3r +2
63 = 3r
r = 21 This means Rachel and Claire are 21 years old. Plug this into the equation so
t = 23 Tim is 23 years old.
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If f(x) = 5x/a, and g(x) = ax/10, what is f(2) + g(5)?
If f(x) = 5x/a, and g(x) = ax/10, what is f(2) + g(5)?
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(20+a2) / 2a is the answer. Find a common denominator to add the 2 fractions after plugging in the values. So we get 10/a + a/2 after plugging in the values. The lowest common denominator would be 2a. Mutiply 10/a by 2/2, and multiply a/2 by a/a. The new equation appears as 20/2a + a2/2a. Add them together to get (20 + a2)/2a.
(20+a2) / 2a is the answer. Find a common denominator to add the 2 fractions after plugging in the values. So we get 10/a + a/2 after plugging in the values. The lowest common denominator would be 2a. Mutiply 10/a by 2/2, and multiply a/2 by a/a. The new equation appears as 20/2a + a2/2a. Add them together to get (20 + a2)/2a.
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(√(4) + √(16))2 = ?
(√(4) + √(16))2 = ?
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If we use order of operations (PEMDAS), the answer is 36.
If we use order of operations (PEMDAS), the answer is 36.
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x * y = –a, a – x = 2a. What is y?
x * y = –a, a – x = 2a. What is y?
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Use the second equation to find x in terms of a. Plug it back in the second equation, that will give you 1 = y.
Use the second equation to find x in terms of a. Plug it back in the second equation, that will give you 1 = y.
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The projected sales of a movie are given by the function s(p) = 3000/(2_p_ + a) where s is the number of movies sold, in thousands; p is the price per movie, in dollars; and a is a constant. If according to projections, 75,000 cartidges will be sold at $15 each, how many movies are sold at $20 each?
The projected sales of a movie are given by the function s(p) = 3000/(2_p_ + a) where s is the number of movies sold, in thousands; p is the price per movie, in dollars; and a is a constant. If according to projections, 75,000 cartidges will be sold at $15 each, how many movies are sold at $20 each?
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You set up the equation to solve for a.
75 = 3000/(2(15) + a)
a = 10
You then set up the formula again for each movie costing $20, s(20)= 3000/(2(20) + 10), and solve for x, the number sold, giving you 60.
You set up the equation to solve for a.
75 = 3000/(2(15) + a)
a = 10
You then set up the formula again for each movie costing $20, s(20)= 3000/(2(20) + 10), and solve for x, the number sold, giving you 60.
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Half of one hundred divided by five and multiplied by one-tenth is .
Half of one hundred divided by five and multiplied by one-tenth is .
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Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.
Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.
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Let x&y be defined as (x – y)xy . What is the value of –1_&_2?
Let x&y be defined as (x – y)xy . What is the value of –1_&_2?
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We are told that x&y = (x – y)xy .
–1&2 = (–1 – 2)(–1)(2) = (–3)–2
To simplify this, we can make use of the property of exponents which states that a– b = 1/(ab ).
(–3)–2 = 1/(–3)2 = 1/9
The answer is 1/9.
We are told that x&y = (x – y)xy .
–1&2 = (–1 – 2)(–1)(2) = (–3)–2
To simplify this, we can make use of the property of exponents which states that a– b = 1/(ab ).
(–3)–2 = 1/(–3)2 = 1/9
The answer is 1/9.
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If ab - bc + d = d2 - c2, then what is the value of a when b is two, c is negative one, and d is zero?
If ab - bc + d = d2 - c2, then what is the value of a when b is two, c is negative one, and d is zero?
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ab - bc + d = d2 - c2
We need to substitute values in for b, c, and d, and then solve the equation for a.
a(2) - 2(-1) + 0 = 02 - (-1)2
2a +2 + 0 = 0 - (1)
2a + 2 = -1
2a = -3
a = -3/2
The answer is -3/2.
ab - bc + d = d2 - c2
We need to substitute values in for b, c, and d, and then solve the equation for a.
a(2) - 2(-1) + 0 = 02 - (-1)2
2a +2 + 0 = 0 - (1)
2a + 2 = -1
2a = -3
a = -3/2
The answer is -3/2.
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If 11x + 4 = 19x – 12, then what is 2x – 4?
If 11x + 4 = 19x – 12, then what is 2x – 4?
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First solve for x. The first equation would simplify as:
16 = 8x
x = 2
If we plug x = 2 into the second expression:
2(2) – 4 = 0
0 is the correct answer.
First solve for x. The first equation would simplify as:
16 = 8x
x = 2
If we plug x = 2 into the second expression:
2(2) – 4 = 0
0 is the correct answer.
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If x = 2 and y = 3, then evaluate 2(x – 3) + 5y2
If x = 2 and y = 3, then evaluate 2(x – 3) + 5y2
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To evaluate an expression we make substitutions into the expression
2(x – 3) + 5y2 becomes 2(2 – 3) + 5 * 32 = –2 + 45 = 43
To evaluate an expression we make substitutions into the expression
2(x – 3) + 5y2 becomes 2(2 – 3) + 5 * 32 = –2 + 45 = 43
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IF 5x3 = 40, then what is the value of 12x – (x/2)?
IF 5x3 = 40, then what is the value of 12x – (x/2)?
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Use the first equation to solve for x, then plug into the 2nd equation to find a value.
5x3 = 40
x3 = 8
x = 2
12(2) – (2/2) = 24 – 1 = 23
Use the first equation to solve for x, then plug into the 2nd equation to find a value.
5x3 = 40
x3 = 8
x = 2
12(2) – (2/2) = 24 – 1 = 23
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Twenty percent of a number, 
, is four greater than the product of that number and six. Which of the following algebraic equations could be used to find 
?
Twenty percent of a number,
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The "is" in the question means "equal," so whatever comes before "is" must be equal to whatever comes after. We will find an expression for the information before "is" and an expression for the information after "is," and then we will set these two expressions equal.
Twenty percent of a number can be represented as 0.2n, because 20% expressed as a decimal is 0.2, and because twenty percent "of" a number means the product of that number and twenty percent.
Four greater than the product of a number and six means that we must first find the product of that number and six, and then increase this value by 4.
The product of a number and six means that we must multiply this number by six, which can be represented by 6n. Increasing 6n by 4 can be modeled by the expression 6n + 4, or 4 + 6n (because of the commutative property of addition).
Setting the two expressions equal gives us 0.2n = 4 + 6n .
The "is" in the question means "equal," so whatever comes before "is" must be equal to whatever comes after. We will find an expression for the information before "is" and an expression for the information after "is," and then we will set these two expressions equal.
Twenty percent of a number can be represented as 0.2n, because 20% expressed as a decimal is 0.2, and because twenty percent "of" a number means the product of that number and twenty percent.
Four greater than the product of a number and six means that we must first find the product of that number and six, and then increase this value by 4.
The product of a number and six means that we must multiply this number by six, which can be represented by 6n. Increasing 6n by 4 can be modeled by the expression 6n + 4, or 4 + 6n (because of the commutative property of addition).
Setting the two expressions equal gives us 0.2n = 4 + 6n .
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An elementary school class consists of 
boys and 
girls. What fraction of the class is female?
An elementary school class consists of
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There are B+G total students in the elementary school class, so G out of B+G are girls.
There are B+G total students in the elementary school class, so G out of B+G are girls.
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A total of 150 million votes were tallied in a presidential election. Votes were cast for either Hillary Clinton, Rand Paul, Al Gore, or Gary Johnson. If Clinton received 3 times the number of votes as Johnson, Paul received 30% of the vote, and Gore receieved 30 million total votes, who received the most votes in the election?
A total of 150 million votes were tallied in a presidential election. Votes were cast for either Hillary Clinton, Rand Paul, Al Gore, or Gary Johnson. If Clinton received 3 times the number of votes as Johnson, Paul received 30% of the vote, and Gore receieved 30 million total votes, who received the most votes in the election?
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There are a few ways to do this problem, but we will focus on the total number of votes method as follows. First, let Clinton = C, Gore = G, Paul = P, and Johnson = J. We know C + G + P + J = 150 million. We also know that C = 3J. Paul received 30% of the vote which is 150,000,000 * .3 = 45 million votes. Gore received 30 million votes. We can now create an equation with individual totals and substitute 3J for Clinton's vote total:
3J + 30 million + 45 million + J = 150 million
4J = 75 million
J = 18.75 million
Then C = 3J = 56.25 million. So Clinton received 56.25 million votes, Paul received 45 million votes, Gore received 30 million votes, and Johnson received 18.75 million votes. The correct answer is Hillary Clinton.
There are a few ways to do this problem, but we will focus on the total number of votes method as follows. First, let Clinton = C, Gore = G, Paul = P, and Johnson = J. We know C + G + P + J = 150 million. We also know that C = 3J. Paul received 30% of the vote which is 150,000,000 * .3 = 45 million votes. Gore received 30 million votes. We can now create an equation with individual totals and substitute 3J for Clinton's vote total:
3J + 30 million + 45 million + J = 150 million
4J = 75 million
J = 18.75 million
Then C = 3J = 56.25 million. So Clinton received 56.25 million votes, Paul received 45 million votes, Gore received 30 million votes, and Johnson received 18.75 million votes. The correct answer is Hillary Clinton.
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Based on the chart, which equation represents the table data?
Based on the chart, which equation represents the table data?
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The easiest way to solve this problem is to guess-and-check the answer choices. The equation that can be used to match the table will be correct.
We can see that the values in the table match the equation 
for each given value. Thus, this must be our answer.
We can also determine certain characteristics from the table itself. For example, as x increases, y(x) decreases. This tells us that there is likely a negative coefficient, which can help narrow down the answer options.
The easiest way to solve this problem is to guess-and-check the answer choices. The equation that can be used to match the table will be correct.
We can see that the values in the table match the equation
We can also determine certain characteristics from the table itself. For example, as x increases, y(x) decreases. This tells us that there is likely a negative coefficient, which can help narrow down the answer options.
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Justin makes 61.9% of his free throws. During the season he had 84 free throw attempts. How many of Jason’s shots did not go in?
Justin makes 61.9% of his free throws. During the season he had 84 free throw attempts. How many of Jason’s shots did not go in?
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Find how many free throws Justin made: 84 x 0.619 = 51.99. Since the problem talks free throws, we round to 52 shots went in. To calculate shots missed:
84 – 52 = 32.
Find how many free throws Justin made: 84 x 0.619 = 51.99. Since the problem talks free throws, we round to 52 shots went in. To calculate shots missed:
84 – 52 = 32.
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If 5x + 30 = 6 – 7x, then x = ?
If 5x + 30 = 6 – 7x, then x = ?
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Combine like terms by subtracting 6 from both sides so: 5x + 24 = –7x. Then subtract 5x from both sides: 24 = –12x. Divide both sides by –12 and x = –2.
Combine like terms by subtracting 6 from both sides so: 5x + 24 = –7x. Then subtract 5x from both sides: 24 = –12x. Divide both sides by –12 and x = –2.
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Drink A is 20% water by weight, and drink B is 35% water by weight. How many fluid ounces of drink A must be added to 80 oz of drink B to have a drink whose final proportion of water is 30%?
Drink A is 20% water by weight, and drink B is 35% water by weight. How many fluid ounces of drink A must be added to 80 oz of drink B to have a drink whose final proportion of water is 30%?
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It's easiest if we convert all percentages to actual oz of water for each step here. As such, 35% of the 80 oz of drink B would have 0.35(80) = 28 oz of water in it.
We can set up an equation that similarly converts each "percentage of a fixed weight of liquid" to ensure that our final weight is equivalent to 30% to the sum of drink A and B. On the left side, the fixed values are the percentages of each drink individually, and on the right side is what the question requires as a fixed percentage of the final weight:
0.2(A) + 0.35(80) = 0.3(A + 80)
0.2A + 28 = 0.3A + 24
A = 40
Solving for A, we get 40 oz of A that must be poured into B. You may plug this back into the equation to check it.
It's easiest if we convert all percentages to actual oz of water for each step here. As such, 35% of the 80 oz of drink B would have 0.35(80) = 28 oz of water in it.
We can set up an equation that similarly converts each "percentage of a fixed weight of liquid" to ensure that our final weight is equivalent to 30% to the sum of drink A and B. On the left side, the fixed values are the percentages of each drink individually, and on the right side is what the question requires as a fixed percentage of the final weight:
0.2(A) + 0.35(80) = 0.3(A + 80)
0.2A + 28 = 0.3A + 24
A = 40
Solving for A, we get 40 oz of A that must be poured into B. You may plug this back into the equation to check it.
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Let x&y = xy – x + y. Which of the following is equal to 5&3 ?
Let x&y = xy – x + y. Which of the following is equal to 5&3 ?
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First, we need to evaluate 5&3.
Since x&y = xy – x + y, we can find 5&3 as follows:
5&3 = 5(3) – 5 + 3
= 15 – 5 + 3
= 13
Now, we need to go through each of the choices and determine which one equals 13.
Let's start with 2&(8&1). Remember that we must first evaluate inside the parantheses.
2&(8&1) = 2&(8(1) – 8 + 1)
= 2&(8 – 8 + 1)
= 2&(1)
= 2(1) – 2 + 1 = 1, which doesn't equal 13.
Next, let's evaluate 4&(3&4).
4&(3&4) = 4&(3(4) – 3 + 4)
= 4&(13) = 4(13) – 4 + 13 = 61.
Next, we can evaulate 8&(2&0).
8&(2&0) = 8&(2(0) – 2 + 0)
= 8&(–2) = 8(–2) – 8 + 2 = –22.
Next, let's find 3&(5&2).
3&(5&2) = 3&(5(2) – 5 + 2)
= 3&(7) = 3(7) – 3 + 7 = 25.
Finally, lets evaluate 11&(0&2).
11&(0&2) = 11&(0(2) – 0 + 2)
= 11&(2) = 11(2) – 11 + 2 = 13.
The answer is 11&(0&2).
First, we need to evaluate 5&3.
Since x&y = xy – x + y, we can find 5&3 as follows:
5&3 = 5(3) – 5 + 3
= 15 – 5 + 3
= 13
Now, we need to go through each of the choices and determine which one equals 13.
Let's start with 2&(8&1). Remember that we must first evaluate inside the parantheses.
2&(8&1) = 2&(8(1) – 8 + 1)
= 2&(8 – 8 + 1)
= 2&(1)
= 2(1) – 2 + 1 = 1, which doesn't equal 13.
Next, let's evaluate 4&(3&4).
4&(3&4) = 4&(3(4) – 3 + 4)
= 4&(13) = 4(13) – 4 + 13 = 61.
Next, we can evaulate 8&(2&0).
8&(2&0) = 8&(2(0) – 2 + 0)
= 8&(–2) = 8(–2) – 8 + 2 = –22.
Next, let's find 3&(5&2).
3&(5&2) = 3&(5(2) – 5 + 2)
= 3&(7) = 3(7) – 3 + 7 = 25.
Finally, lets evaluate 11&(0&2).
11&(0&2) = 11&(0(2) – 0 + 2)
= 11&(2) = 11(2) – 11 + 2 = 13.
The answer is 11&(0&2).
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