Algebraic Equations - Math

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Question

The square root of the sum of twice a number and five is equal to the square of the difference between that number and two. Which of the following equations could be used to find the number, represented by n?

Answer

We need to translate this word sentence into algebra: The square root of the sum of twice a number and five is equal to the square of the difference between that number and two.

The words "is equal to" denote the equal sign. Everything before it will be to the left of the equal sign, and everything after will be on the right.

The first half of the sentence says "The square root of the sum of twice a number and five." We can represent twice the number as 2n. The sum of twice the number and five can be represented by 2n + 5, because "sum" denotes addition. We need to take the square root of the quantity 2n + 5, which we can write as $\sqrt{2n+5}$ . The left side of the equation will be $\sqrt{2n+5}$ .

On the right side, we need to find the square of the difference between n and 2. "Difference" tells us to subtract, so we can represent this as n - 2. If we take the square of this quantity, we can represent this as $(n-2)^2$ . The right side of the equation is $(n-2)^2$ .

Putting both sides together with the equal sign, the entire equation is $\sqrt{2n+5}$ $=(n-2)^2$ .

The answer is $\sqrt{2n+5}$ $=(n-2)^2$ .

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Question

Solve for X in the following equation:

Answer

To solve for X, isolate X on one side of the equation. In this case, we are required to subtract one number from X. When we add that number to both sides of the equation, we isolate X and obtain our answer.

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Question

Which equation can be used to solve the following problem?

Jason is planning a birthday party at a bowling alley. It costs to rent the alley and per attendee. What is the total cost () of the party in terms of , where equals the total number of attendees?

Answer

The cost per attendee, , must be multiplied by the total number of attendees and then added to the base cost, , to obtain the total cost.

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Question

A family has 7 animals. Write an equation for the number of cats, , in terms of the number of dogs, . Assume the family has only cats and dogs.

Answer

In order to sovle for , subtract from both sides:

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Question

Sally went shopping and bought two shirts. One shirt cost $15, and she paid a total of $27 for both shirts. What was the cost of the other shirt?

Answer

To solve this word problem, set up an algebraic equation, putting in for the unknown value of the second shirt. The equation is because we know the cost of the two shirts will total .

To solve for , subtract from each side. This results in .

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Question

Rojo Salsa is on sale at a price of for jars of ounces each. Verde Salsa is on sale at a price of for jars of ounces each. Which of the following statements is true?

Answer

The statement "A jar of Verde Salsa costs more than a jar of Rojo Salsa" can be tested by comparing the price per jar of each salsa.

$\frac{$9}{3 \ jars \ Rojo} = $3 \ per \ jar$ versus $\frac{$8}{4 \ jars \ Verde} = $2 \ per \ jar$

The statement is false since the price of Rojo per jar is greater.

The remaining statements above can all be proven true or false by finding the price per ounce of each salsa.

Rojo Salsa is on sale at a price of for jars of ounces each. The following operations can be used to determine the cost of Rojo Salsa per ounce:

$3 \ jars \times 12 \ ounces \ per \ jar =36 \ ounces \ sold \ per \ $9$

$\rightarrow $9 = 36 \ ounces \rightarrow $ \frac{9}{36} = 1 \ ounce \rightarrow $ \frac{1}{4} \ per \ ounce$

$\rightarrow $0.25 \ per \ ounce$ for Rojo Salsa.

Verde Salsa is on sale at a price of for jars of ounces each. The following operations can be used to determine the cost of Verde Salsa per ounce.

$4 \ jars \times 8 \ ounces \ per \ jar =32 \ ounces \ sold \ per \ $8$

$\rightarrow $8 = 32 \ ounces \rightarrow $ \frac{8}{32} = 1 \ ounce \rightarrow $ \frac{1}{4} \ per \ ounce$

$\rightarrow $0.25 \ per \ ounce$ for Verde Salsa.

The only true statement is "An ounce of Rojo Salsa is the same price as an ounce of Verde Salsa."

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Question

Barbara lives miles from the beach, and her friend Josef lives miles from the beach. If Barbara and Joe leave their homes at the same time, and Barbara drives miles per hour, how fast will Joe need to drive to arrive at the beach at the exact same time as Barbara?

Answer

To find the speed (or rate) that Josef will need to travel, we can use the equation $distance = rate\times time$ ().

This equation cannot be used for Josef yet, since only the distance traveled is known and not the time in which he will need to make the trip.

To find the time it takes Barbara to make the trip, use the same equation to solve for , where the distance is the length of Barbara's trip. Note that we express miles per hour as a fraction that represents the ratio of miles to hour.

$16 \ miles = \frac{60 \ miles}{1 \ hour} \times t \ hours$

Multiply both sides of the equation by the reciprocal of the rate. Note that the unit of "miles" cancels out, leaving only the unit "hours" (time). The result will be expressed as a fraction of a single hour.

$(16 \ miles) \times \frac{1 \ hour}{60 \ miles} = \left ( \frac{60 \ miles}{1 \ hour} \times t \ hours)\right) \times \frac{1 \ hour}{60 \ miles}$

$\rightarrow \frac{16}{60} \ hours = t\rightarrow \frac{4}{15} \ hours = t$

The amount of time it takes Barbara to get to the beach must be the same amount of time it takes Joseph to get to the beach.

Therefore, we can use this new value of and the equation to find the rate Josef will need to travel for his mile trip.

$20 \ miles = (r \ miles \ per \ hour) \times (\frac{4}{15} \ hours)$

Multiply both sides by the reciprocal of time (a fraction) to isolate the rate.

$20 \ miles \times \left ( \frac{15}{4} \ hours\right) = \left( \frac{r \ miles}{1 \ hour}\right) \times \frac{4}{15} \ hours \times \frac{15}{4} \ hours$ $\rightarrow \frac{300 \ miles}{4 \ hours} = \frac{r \ miles}{1 \ hour} \rightarrow 75 \ miles \ per \ hour = r$

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Question

Sarah sells necklaces for each. She sells necklaces in a month at this price. If she applied a % discount to the price of her necklaces, she would sell an additional necklaces in a month. How much additional money would Sarah make in sales if she sold her necklaces with the % discount for a month?

Answer

To find how much additional money Sarah would make by applying the discount, find the difference between her earnings in a normal month and a month where the discount is applied. In a normal month, multiply the normal price by the normal quantity sold to find the normal earnings:

$40 \ (necklaces \ sold) \times $50 \ (price \ per \ necklace) = $2000 \ earned$

To find earnings at the discounted price, first calculate how much each necklace will cost with the % discount. To do this, subtract the amount discounted (calculated by percent as a fraction of multiplied by the original price) from the original price.

$$50 - (\frac{10}{100} \times$50) = $50 - $5 = $45$

If necklaces are sold at the new discounted price of each, multiply these together to find the total earnings with the discount.

$$45 \times 50 = $2250$

Finally, subtract the earnings without the discount from the earnings with the discount to find the additional money made by applying the discount.

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Question

A shirt costs $12 after a 15% discount. What was the original price of the shirt?

Answer

Convert 15% to a decimal.

Let the original price equal $\small x$ . The discount will be 15% of $\small x$ . Subtracting the discount from the original price will equal the amount paid, $12.

Using this equation, we can solve for $\small x$ .

$\frac{0.85x}{0.85}=\frac{12}{0.85}$

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Question

A farmer has units of fence. If he uses this to build a square fence, what will be the length of each side?

Answer

If this is a square fence, then each of the four sides will be equal.

The fence in question will become the perimeter of that square.

Since when working with a square, for this problem .

$\frac{100}{4}=s$

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Question

The area of a rectangle is . How many whole by rectangles can fit inside of this larger rectangle?

Answer

First we need to find the area of the smaller rectangle.

Now to find out how many can fit, we divide the total area by the smaller area.

$\frac{200ft^2}{15ft^2}=x$

$13\tfrac{1}{3}=x$

However, the problem is asking how many WHOLE rectangles can fit. Therefore only can fit.

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Question

40% of Eric's socks are black, and the rest are white. If he has 6 pairs of black socks, how many pairs of white socks does he have?

Answer

First, use the information to find the total number of socks Eric has by setting up a proportion:

$\frac{40}{100}=\frac{6}{x}$ , where is the number of total socks.

Once you cross multiply and divide, you will find that .

To find the number of white socks Eric has, subtract the number of black socks from the total number of socks:

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Question

A chemistry student has of a acid solution. She needs a acid solution for an experiement. How much pure water should she add?

Answer

pure water and pure acid

In general, mixture problems have the form:

$V_{1}P_{1} + V_{2}P_{2} = V_{f}P_{f}$ , where volume and percent

The equation to solve becomes:

$\frac{0.50(0.75-0.50)}{0.50}=x$

Then the solution is .

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Question

Sarah is having a birthday party. She wants to invite friends. Each invitation costs $50\cent$ , and each stamp costs $45\cent$ . If she plans to mail all of the invitations, how much change should she get back if she pays with a bill?

Answer

The cost of mailing each invitation is .

To invite everyone, it will cost $0.95\cdot 18=17.10$ .

Then the change is .

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Question

What number is of ?

Answer

Look for the verbal cues in the question. IS translates to "equals" and OF translates to multiplication.

Thus the equation to solve becomes:

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Question

What percent of is ?

Answer

Look for the verbal cues in the question. IS represents "equals" and OF represents multiplication.

Then the equation to solve becomes:

$P\cdot 75=15$ so or

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Question

Joy earns a commission on each house she sells. How much money does she make when she sells two houses for each and another house for ?

Answer

In general, a commission problem has the form:

$$; sales \cdot %; commission = $ ; commission$

So the commission is:

$213,000\cdot 0.05=$10,650$

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Question

Dean is counting his money. He has five more quarters than nickles and four less nickles than dimes. He has . How many coins does he have?

Answer

Money problems generally consist of two parts: total number of coins and total value of money.

Let number of nickles, number of quarters, and number of dimes.

So the equation to solve becomes:

or

so

Thus, , so there are three nickles, eight quarters and seven dimes for a total of coins.

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Question

Edward is making cookies. The full recipe makes cookies. He decides to cut the recipe by four. How many dozens of cookies will he make?

Answer

Cut the recipe by four: $144\div 4=36$ cookies; however, the question asks for how many DOZENS of cookies are made, so the correct answer is dozen cookies.

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Question

Golden Sunshine paint is made by mixing one part red paint and three parts yellow paint. How many gallons of yellow paint should be mixed with three quarts of red paint?

Answer

Use proportions to solve this problem.

Let quarts of yellow paint.

$\frac{3; yellow}{1; red}= \frac{x; qt ; yellow; paint}{3; qt ; red; paint}$

By cross-multiplying we get .

Since , we actually need .

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