Word Problems with Two Unknowns - Pre-Algebra

Step 1: Megan and Kelly's total hours worked needs to add up to 60, and Kelly worked two times as long as Megan. We can put this into a formula:

Step 2: substitute 2m for k and add the variables

Step 3: isolate m

Step 4: Now that we know Megan worked 20 hours (m=20), we can multiply her hours worked by 2 to find out how long Kelly worked.

Step 5: check to make sure Megan and Kelly's hours add up to 60

To solve this, we can set each of their ages as a variable. Let's say Michael's age is x.

We know Tom is 4 years older than Michael, so Tom's age is x+4.

We also know that their combined age is 20, so if we add both of their ages, we should get 20.

x + (x+4) = 20

2x+4 = 20

2x=16

x=8.

So Michael's age is 8, and Tom is 12.

First, set up the equations. Their base pay is constant per hour, so the variable is the number of calculators sold multiplied by their comission rate, so the earning equations will be :

and

These are Sara's earnings and Jamie's earnings respectively, with representing calculators sold and representing earnings.

Because we are trying to find the number of calculators sold when both women have equal earnings, we can set as equal to , then substitute this value for in Jamie's equation, giving the equation:

Next, subtract and from both sides to get:

Which simplifies to

We can solve this problem by setting up an algebra equation. We know Jamarcus has twenty-one coins, but we don't know how many of each he has. That usually means we need a variable. Since we don't know how many dimes he has, let's label d as the number of dimes. If we want to find the number of quarters, we would subtract the number of dimes from 21, and the number we get would be the number of quarters. Therefore, if Jamarcus hs dimes, he must have quarters. We can double check ourselves. If we add the number of dimes and quarters, we get 21.

Now, the only other piece of information we have is that together all 21 coins add up to $4.20. At first that might not seem too helpful, but it actually allows us to solve the problem. We know that every dime is worth 10 cents, so every dime Jamarcus has contributes 10 cents towards his $4.20 total. Furthermore, each quarter contributes 25 cents to his total. Since Jamarcus has dimes and each is worth 10 cents, the total value of his dimes is just . Furthermore, since Jamarcus has quarters each worth 25 cents, the total value of all of his quarters is . The sum of these two totals should equal the grand total of $4.20, or 420 cents. We can write this as the following equation.

Next we use the distributive property to simplify, multiplying the 25 by both the 21 and the .

Simplifying further we get

We then want to combine like terms (the _d_s)

We then want all of our variables on one side and all of our constants on the other, which we can accomplish by subtracting 525 from both sides.

which gives

To solve for , we now simply need to divide both sides by .

which gives

That means Jamarcus has 7 dimes. If we remember that he had 21 coins in all, that leaves 14 quarters. Jamarcus has 7 dimes and 14 quarters.

We can double check ourselves. Seven dimes would total $0.70, and 14 quarters would total $3.50, bringing the grand total to the correct value of $4.20.

We can solve this problem easily by using elementary algebra. We do not know either of the two numbers, so for the meantime let's label the second number as . Since the first number is 18 more than the second, we can express the first number as . Since the two numbers add up to 128, we can write this fact as an equation.

We can then combine like terms (our variables), giving us

We then want all of our constant terms on the right side, which we can accomplish by subtracting 18 from both sides.

The last step to solving the equation is to divide both sides by 2.

Therefore, our second number is 55. Since our first number is 18 more than that, it must equal 73. Double checking, we can confirm that the sum of 55 and 73 is indeed 128.

We need to translate the English words into a mathematical statement.

Product means multiply.

Difference means subtraction.

Is means equals.

Product of 3 and s is 3s.

Difference of 12 and 7 is 12 - 5.

Therefore, the equation becomes,

.

Write two equations to represent the scenario. There are two equations and two unknowns.

A total of 14 coins composed of dimes, , and nickels, . Write the first equation.

Nickels are 5 cents, and dimes are 10 cents. The total is 80 cents. Write the second equation.

Multiply the second equation by 10 and use the elimination method to cancel out the dimes variable.

Subtract the first and the new second equation to eliminate and solve for .

Divide by on both sides.

There are 12 nickels.

Substitute this into the first equation to find the number of dimes.

There are 2 dimes and 12 nickels.

Given the information, we have 2 equations. We know each bag of carrots is $1.50 and each banana is $0.25. We also know the total amount we spend is $6.50. So, we can write the equation

where x is the number of bags of carrots and y is the number of bananas.

We also know the total number of items we purchased is 11. We can write the equation as

where x is number of bags of carrots and y is the number of bananas.

To solve, we will solve for one variable in one equation and substitute it into the other equation. So,

Now, we can substitute the value of y into the first equation. We get,

We distribute.

We combine like terms.

We solve for x by getting x alone.

Therefore, the number of bags of carrots we bought is 3. To find the number of bananas, we simply substitute x into the equation.

Therefore, the number of bananas we bought is 8.

So we bought 3 bags of carrots and 8 bananas.

Step 1: find the number of cupcakes ordered by adding up all of the people at the party and then multiplying that number by the 2 cupcakes ordered per person.

Step 2: to find the number of cupcakes each person can have, take the number of cupcakes and divide it by the number of guests, including Jamal.

Step 1: Megan and Kelly's total hours worked needs to add up to 60, and Kelly worked two times as long as Megan. We can put this into a formula:

Step 2: substitute 2m for k and add the variables

Step 3: isolate m

Step 4: Now that we know Megan worked 20 hours (m=20), we can multiply her hours worked by 2 to find out how long Kelly worked.

Step 5: check to make sure Megan and Kelly's hours add up to 60

To solve this, we can set each of their ages as a variable. Let's say Michael's age is x.

We know Tom is 4 years older than Michael, so Tom's age is x+4.

We also know that their combined age is 20, so if we add both of their ages, we should get 20.

x + (x+4) = 20

2x+4 = 20

2x=16

x=8.

So Michael's age is 8, and Tom is 12.

First, set up the equations. Their base pay is constant per hour, so the variable is the number of calculators sold multiplied by their comission rate, so the earning equations will be :

and

These are Sara's earnings and Jamie's earnings respectively, with representing calculators sold and representing earnings.

Because we are trying to find the number of calculators sold when both women have equal earnings, we can set as equal to , then substitute this value for in Jamie's equation, giving the equation:

Next, subtract and from both sides to get:

Which simplifies to

We can solve this problem by setting up an algebra equation. We know Jamarcus has twenty-one coins, but we don't know how many of each he has. That usually means we need a variable. Since we don't know how many dimes he has, let's label d as the number of dimes. If we want to find the number of quarters, we would subtract the number of dimes from 21, and the number we get would be the number of quarters. Therefore, if Jamarcus hs dimes, he must have quarters. We can double check ourselves. If we add the number of dimes and quarters, we get 21.

Now, the only other piece of information we have is that together all 21 coins add up to $4.20. At first that might not seem too helpful, but it actually allows us to solve the problem. We know that every dime is worth 10 cents, so every dime Jamarcus has contributes 10 cents towards his $4.20 total. Furthermore, each quarter contributes 25 cents to his total. Since Jamarcus has dimes and each is worth 10 cents, the total value of his dimes is just . Furthermore, since Jamarcus has quarters each worth 25 cents, the total value of all of his quarters is . The sum of these two totals should equal the grand total of $4.20, or 420 cents. We can write this as the following equation.

Next we use the distributive property to simplify, multiplying the 25 by both the 21 and the .

Simplifying further we get

We then want to combine like terms (the _d_s)

We then want all of our variables on one side and all of our constants on the other, which we can accomplish by subtracting 525 from both sides.

which gives

To solve for , we now simply need to divide both sides by .

which gives

That means Jamarcus has 7 dimes. If we remember that he had 21 coins in all, that leaves 14 quarters. Jamarcus has 7 dimes and 14 quarters.

We can double check ourselves. Seven dimes would total $0.70, and 14 quarters would total $3.50, bringing the grand total to the correct value of $4.20.

We can solve this problem easily by using elementary algebra. We do not know either of the two numbers, so for the meantime let's label the second number as . Since the first number is 18 more than the second, we can express the first number as . Since the two numbers add up to 128, we can write this fact as an equation.

We can then combine like terms (our variables), giving us

We then want all of our constant terms on the right side, which we can accomplish by subtracting 18 from both sides.

The last step to solving the equation is to divide both sides by 2.

Therefore, our second number is 55. Since our first number is 18 more than that, it must equal 73. Double checking, we can confirm that the sum of 55 and 73 is indeed 128.

We need to translate the English words into a mathematical statement.

Product means multiply.

Difference means subtraction.

Is means equals.

Product of 3 and s is 3s.

Difference of 12 and 7 is 12 - 5.

Therefore, the equation becomes,

.

Write two equations to represent the scenario. There are two equations and two unknowns.

A total of 14 coins composed of dimes, , and nickels, . Write the first equation.

Nickels are 5 cents, and dimes are 10 cents. The total is 80 cents. Write the second equation.

Multiply the second equation by 10 and use the elimination method to cancel out the dimes variable.

Subtract the first and the new second equation to eliminate and solve for .

Divide by on both sides.

There are 12 nickels.

Substitute this into the first equation to find the number of dimes.

There are 2 dimes and 12 nickels.

Given the information, we have 2 equations. We know each bag of carrots is $1.50 and each banana is $0.25. We also know the total amount we spend is $6.50. So, we can write the equation

where x is the number of bags of carrots and y is the number of bananas.

We also know the total number of items we purchased is 11. We can write the equation as

where x is number of bags of carrots and y is the number of bananas.

To solve, we will solve for one variable in one equation and substitute it into the other equation. So,

Now, we can substitute the value of y into the first equation. We get,

We distribute.

We combine like terms.

We solve for x by getting x alone.

Therefore, the number of bags of carrots we bought is 3. To find the number of bananas, we simply substitute x into the equation.

Therefore, the number of bananas we bought is 8.

So we bought 3 bags of carrots and 8 bananas.

Step 1: find the number of cupcakes ordered by adding up all of the people at the party and then multiplying that number by the 2 cupcakes ordered per person.

Step 2: to find the number of cupcakes each person can have, take the number of cupcakes and divide it by the number of guests, including Jamal.

Step 1: Megan and Kelly's total hours worked needs to add up to 60, and Kelly worked two times as long as Megan. We can put this into a formula:

Step 2: substitute 2m for k and add the variables

Step 3: isolate m

Step 4: Now that we know Megan worked 20 hours (m=20), we can multiply her hours worked by 2 to find out how long Kelly worked.

Step 5: check to make sure Megan and Kelly's hours add up to 60