Single-Variable Algebra - SAT Math

Card 0 of 760

Question

If Jane buys four cans of soda at $1.56 each and pays $6.93 total, what is the percentage of the sales tax?

Answer

How to calculate the amount of sales tax?

Convert tax percentage into a decimal by moving the decimal point two spaces to the left.

Multiple the pre-tax value by the newly calculated decimal value in order to find the cost of the sales tax.

Add the sales tax value to the pre-tax value to calculate the total cost.

Calculating sales tax at time of purchase:

In order to calculate the sales tax of an item, we need to first multiply the pre-tax cost of the item by the sales tax percentage after it has been converted into a decimal. Once the sales tax has been calculated it needs to be added to the pre-tax value in order to find the total cost of the item. Let's start by working with an example. If a magazine costs $2.35 and has a 6% sales tax, then what is the total cost of the item. First, we need to convert the sales tax percentage into a decimal by moving the point two spaces to the left.

$6% \rightarrow 0.06$

Now, we need to multiply the pre-tax cost of this item by this value in order to calculate the sales tax cost.

$\textup{Sales tax}=0.06\times$2.35$

$\textup{Sales tax}=$0.141$

Round to two decimal places since our total is in dollars and cents.

$\textup{Sales tax}=$0.14$

Last, add this value to the pre-tax value of the item to find the total cost.

$\textup{Total cost}=\textup{Pre-tax value}+\textup{Sales tax}$

$\textup{Total cost}=$2.35+$0.14$

$\textup{Total cost}=$2.49$

Calculating the sales tax percentage of a total:

If we are given the total cost of an item or group of items and the pre-tax cost of the good(s), then we can calculate the sales tax percentage of the total cost. First, we need to subtract the pre-tax value from the total cost of the purchase. Next, we need to create a ratio of the sales tax to the pre-tax cost off the items. Last, we need to create a proportion where the pre-tax cost is related to 100% and solve for the percentage of the sales tax. Let's start by working through an example. If a person pays $245.64 for groceries that cost $220.00 pre tax, then what is the sales tax percentage for the items.

First, subtract the pre-tax value from the total cost of the items to find the sales tax cost.

$\textup{Sales tax}=\textup{Total cost}-\textup{Pre-tax value}$

$\textup{Sales tax}=$245.64-$220.00$

$\textup{Sales tax}=$25.64$

Next, create a ratio of the sales tax to the pre-tax cost of the items.

$\frac{\textup{Sales tax}}{\ \textup{Pre-tax value}} = \frac{$25.64}{$220.00}$

Last, create a proportion where the pre-tax value is proportional to 100% and solve for the percentage of sales tax.

$\frac{$25.64}{$220.00}=\frac{\textup{Sales tax percentage}}{100%}$

Cross multiply and solve.

$$220.00\times\textup{Sales tax percentage}=$25.64\times 100%$

$$220.00\times\textup{Sales tax percentage}=$2564.00$

Isolate the sales tax percentage to the left side of the equation by dividing each side by the pre-tax value.

$\frac{$220.00\times\textup{Sales tax percentage}}{$220.00}=\frac{$2564.00}{$220.00}$

$\textup{Sales tax percentage}=11.6545455%$

Round to two decimal places since our answer is in dollars and cents.

$\textup{Sales tax percentage}=11.65%$

Last, we can check this answer by calculating the sales tax percentage of the total as seen previously.

First, we need to convert the sales tax percentage into a decimal by moving the point two spaces to the left.

$11.6545455% \rightarrow 0.116545455$

Now, we need to multiply the pre-tax cost of this item by this value in order to calculate the sales tax cost.

$\textup{Sales tax}=0.116545455\times$220.00$

$\textup{Sales tax}=$25.6400001$

Round to two decimal places since our total is in dollars and cents.

$\textup{Sales tax}=$25.64$

Last, add this value to the pre-tax value of the item to find the total cost.

$\textup{Total cost}=\textup{Pre-tax value}+\textup{Sales tax}$

$\textup{Total cost}=$220.00+$25.64$

$\textup{Total cost}=$245.64$

Our answers check out; therefore they are correct. Now, let's use this information to solve the given problem.

Solution:

If each soda cost $1.56 and Jane bought four, the total for the sodas was:

$1.56\cdot 4=6.24$

And she paid $6.93

$\frac{6.93}{6.24}=1.11$

The sales tax was 11%.

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Question

Which of the following phrases can be written as the algebraic expression $\left (8- x \right )^{2}$ ?

Answer

$\left (8- x \right )^{2}$ is the square of the expression ; is the difference of eight and a number.

Subsequently, $\left (8- x \right )^{2}$ is "the square of the difference of eight and a number".

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Question

If Bob fills a bucket with 9 logs that weigh 1.22 pounds each and the total weight of the bucket and logs 13.3 pounds, what percentage of the total weight is the bucket?

Answer

This question involves muliple steps to attain the correct answer.

To solve for the weight of the bucket (we will let 'x' represent this) the following equation can be set up:

To isolate x, subtract 9 * 1.22 (which is equal to 10.98) from both sides. We find x to be equal to 2.32.

The question asks us for the percentage of the total weight, so to find this, we must divide 2.32 by the total weight and multiple our answer by 100.

$\frac{2.32}{13.3}*100 = 17.44$

Rounded to two decimal points of accuracy, the final answer is 17.44%.

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Question

Which of the following phrases can be written as the algebraic expression $\left (12-a \right )^{3}$ ?

Answer

$\left (12-a \right )^{3}$ is the cube of , which is the difference of twelve and a number. Therefore, $\left (12-a \right )^{3}$ is "the cube of the difference of twelve and a number."

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Question

Which of the following phrases can be represented by the algebraic expression $\frac{1}{x} - 9$ ?

Answer

$\frac{1}{x} - 9$ is nine less than $\frac{1}{x}$ , which is the reciprocal of a number. Therefore, $\frac{1}{x} - 9$ is "nine less than the reciprocal of a number. "

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Question

Which of the following phrases can be represented by the algebraic expression

$\sqrt{y - 18}$ ?

Answer

$\sqrt{y - 18}$ is the square root of ,which is the difference of a number and eighteen; therefore, $\sqrt{y - 18}$ is "the square root of the difference of a number and eighteen".

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Question

Jim sells apples for per pound and oranges for per pound. What is a formula to find the revenue for pounds of apples and pounds of oranges?

Answer

Since you are solving for , that goes on one side of the equation. and 's revenues are based on how much they sell for per pound.

So for every pound of apples they make dollars or .

For every pound of oranges they make dollars or .

To put it all together, you get

.

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Question

Sam currently has five ducks. He will receive 3 new ducks every two months. How many individual ducks must he have after 7 months?

Answer

Sam starts out with five ducks. He receives 3 ducks after every two months.

where $m = \textup{every 2 months}$ .

After the first 2 months, he will have 8 ducks.

After the next 2 months, he will have 11 ducks.

After the next 2 months, he will then have 14 ducks.

However, Sam will not receive 3 ducks until the 8th month. This means that he will only have 14 ducks in possession by the 7th month.

The answer is:

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Question

$\small \frac{x^2-5x-6}{x+1}$

For all values , which of the following is equivalent to the expression above?

Answer

First, factor the numerator. We need factors that multiply to $\small -6$ and add to $\small -5$ .

$\small 1*-6=-6\ \text{and}\ 1+(-6)=-5$

$\small x^2-5x-6=(x-6)(x+1)$

We can plug the factored terms into the original expression.

$\small \frac{x^2-5x-6}{x+1}=\frac{(x-6)(x+1)}{(x+1)}$

Note that $\small (x+1)$ appears in both the numerator and the denominator. This allows us to cancel the terms.

$\frac{(x-6)(x+1)}{(x+1)}=(x-6)$

This is our final answer.

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Question

Simplify the expression.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

Answer

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}$

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

$\small \frac{(x^2y^2)(x^3y^4z)}{x^4y^3z^4}=\frac{x^2y^2x^3y^4z}{x^4y^3z^4}$

To combine like terms in the numerator, we add their exponents.

$\small \frac{x^5y^6z}{x^4y^3z^4}$

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

$x^{5-4}y^{6-3}z^{1-4}=xy^3z^{-3}$

Remember that any negative exponents stay in the denominator.

$\frac{xy^3}{z^3}$

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Question

Give the value of that makes the polynomial $9x^{2}+ Nx + 100$ the square of a linear binomial.

Answer

A quadratic trinomial is a perfect square if and only if takes the form

$A^{2} \pm 2AB + B^{2}$ for some values of and .

$9x^{2}+ Nx + 100 = (3x)^{2} + Nx + 10^{2}$ , so

and .

For $9x^{2}+ Nx + 100$ to be a perfect square, it must hold that

$Nx = 2AB = 2 \cdot 3x \cdot 10 = 60x$ ,

so . This is the correct choice.

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Question

How many of the following are prime factors of $81y^{4}- 1$ ?

I) $9y^{2} + 1$

II) $3y^{2} + 1$

III)

IV)

Answer

Factor $y^{4}- 81$ all the way to its prime factorization.

$y^{4}- 81$ can be factored as the difference of two perfect square terms as follows:

$y^{4}- 81$

$=\left ( 9y^{2} \right )^{2} - 1^{2}$

$=\left (9 y^{2} + 1 \right )\left ( 9y^{2} - 1 \right )$

$9y^{2} + 1$ is a factor, and, as the sum of squares, it is a prime. $9y^{2} - 1$ is also a factor, but it is not a prime factor - it can be factored as the difference of two perfect square terms. We continue:

$\left (9 y^{2} + 1 \right )\left ( 9y^{2} - 1 \right )$

$=\left ( 9y^{2} + 1 \right )\left [ \left (3y \right ) ^{2} - 1^{2} \right ]$

$=\left ( 9y^{2} + 1 \right )\left (3y+1 \right )\left (3 y-1\right )$

Therefore, of the given four choices, only $3y^{2} + 1$ is not a factor, so the correct response is three.

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Question

Factor:

$27Q^{3}- 270Q^{2}+900Q -1,000$

Answer

$27Q^{3}- 270Q^{2}+900Q -1,000$

$=\left (3Q \right )^{3}- 3 \cdot 90Q^{2}+3\cdot 300Q -10^{3}$

$=\left (3Q \right )^{3}- 3 \cdot 9Q^{2}\cdot 10+3\cdot 3Q \cdot 10^{2} -10^{3}$

$=\left (3Q \right )^{3}- 3 \cdot \left (3Q \right )^{2}\cdot 10+3\cdot 3Q \cdot 10^{2} -10^{3}$

This can be factored out as the cube of a difference, where :

$A^{3}- 3A^{2}B+3AB^{2} -B^{3} = (A - B)^{3}$

Therefore,

$27Q^{3}- 270Q^{2}+900Q -1,000$

$=\left (3Q \right )^{3}- 3 \cdot \left (3Q \right )^{2}\cdot 10+3\cdot 3Q \cdot 10^{2} -10^{3}$

$= (3Q - 10)^{3}$

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Question

Simplify:

$\frac{(x^5y^2z^3)(x^2yz)}{x^4y^{-3}z^3}$

Answer

To simplify, we begin by simplifying the numerator. When muliplying like bases with different exponents, their exponents are added.

For x: $\small 4+3=7$

For y: $\small 2+1=3$

For z: $\small 3+1=4$

The numerator is now .

When dividing like bases, their exponents are subtracted.

For x: $\small 7-4=3$

For y: $\small 3-(-3)=6$

For z: $\small 4-3=1$

Thus, our answer is .

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Question

Which of the following linear binomials is a factor of the polynomial $x^{3} -34x- 240$ ?

Answer

By the factor theorem, a polynomial is divisible by the linear binomial if and only if . We can use this fact to test each of the binomials by evaluating the dividend for the appropriate value of .

: Evaluate the polynomial at :

$x^{3} -34x- 240 = (-4)^{3} -34(-4)- 240 = -64 +136 - 240 = -170 eq 0$

: Evaluate the polynomial at :

$x^{3} -34x- 240 = (-6)^{3} -34(-6)- 240 = -216+204 -240= -252 eq 0$

: Evaluate the polynomial at :

$x^{3} -34x- 240 =6^{3} -34(6)- 240 = 216-204 -240= -228 eq 0$

: Evaluate the polynomial at :

$x^{3} -34x- 240 =8^{3} -34(8)- 240 = 512-272 -240= 0$

: Evaluate the polynomial at :

$x^{3} -34x- 240 =10^{3} -34(10)- 240 = 1,000-340 -240= 420$

The dividend assumes the value of 0 at , so of the choices given, is the factor.

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Question

What is increased by 40%?

Answer

A number increased by 40% is equivalent to 100% of the number plus 40% of the number. This is taking 140% of the number, or, equivalently, multiplying it by 1.4.

Therefore, increased by 40% is 1.4 times this, or

$1.4 \left (3t+ 7 \right ) = 1.4 \cdot 3t + 1.4 \cdot 7 = 4.2 t + 9.8$

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Question

Simplify the expression:

$\frac{x^2+2x-8}{x+4}$

Answer

To solve this problem, we first need to factor the numerator. We are looking for two numbers that multiply to equal -8 and sum to equal 2.

Now, we can write out our expression in fraction form.

$\frac{(x+4)(x-2)}{x+4}$

Since we have the like term in the numerator and denominator, we can cancel them out of our expression.

$\frac{(x+4)(x-2)}{x+4}= x-2$

Thus, our answer is .

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Question

Which of the following is a prime factor of $x^{6} - 1$ ?

Answer

$x^{6} - 1$ is the difference of two squares:

$x^{6} - 1 =\left ( x^{3} \right ) ^{2}- 1$

As such, it can be factored as follows:

$x^{6} - 1 =\left ( x^{3} \right ) ^{2}- 1 = \left ( x^{3} + 1 \right ) \left ( x^{3} - 1 \right )$

The first factor is the sum of cubes and the second is the difference of cubes; each can be factored further:

$x^{3} +1 = (x+1) \left ( x^{2}-x \cdot 1 + 1^{2} \right ) = (x+1) \left ( x^{2} - x + 1 \right )$

$x^{3} - 1 = (x-1) \left ( x^{2} + x \cdot 1 + 1^{2} \right ) = (x-1) \left ( x^{2} + x + 1 \right )$

Therefore,

$x^{6} - 1 = \left ( x^{3} + 1 \right ) \left ( x^{3} - 1 \right ) = (x+1) \left ( x^{2} - x + 1 \right ) (x-1) \left ( x^{2} + x + 1 \right )$

Of the choices, $x^{2} + x + 1$ appears in the prime factorization and is therefore the correct choice.

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Question

Decrease by 20%. Which of the following will this be equal to?

Answer

A number decreased by 20% is equivalent to 100% of the number minus 20% of the number. This is taking 80% of the number, or, equivalently, multiplying it by 0.8.

Therefore, decreased by 20% is 0.8 times this, or

$0.8 \left (9x + 4y \right ) = 0.8 \cdot 9x + 0.8 \cdot 4y = 7.2x + 3.2y$

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Question

Divide:

$\left ( 12 x ^{4} + 16x^{3}+ 64x^{2}+8 x \right ) \div 8x^{2}$

Answer

Divide termwise:

$\left ( 12 x ^{4} + 16x^{3}+ 64x^{2}+8 x \right ) \div 8x^{2}$

$=\frac{ 12 x ^{4} + 16x^{3}+ 64x^{2}+8 x }{ 8x^{2}}$

$=\frac{ 12 x ^{4} }{ 8x^{2}} + \frac{ 16x^{3} }{ 8x^{2}} + \frac{ 64x^{2} }{ 8x^{2}} +\frac{ 8 x }{ 8x^{2}}$

$=\frac{ 12 }{ 8 }x ^{4-2} + \frac{ 16 }{ 8}x^{3-2} + \frac{ 64 }{ 8 } +\frac{ 8 }{ 8} \cdot \frac{1}{x^{2-1}}$

$=\frac{ 3 }{ 2 }x ^{2} + 2x + 8 + \frac{1}{x}$

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