3.25x + 12.25 > 26.2

Question

Darius has a weed sprayer that will hold 26.2 cups of fluid. The broad-leaf herbicide that he uses comes in 12.25 cup packs that dissolve in water and it is recommended that 1 pack is used for every 1,000 square feet of yard being sprayed. The herbicide package also recommends adding in 3.25 cups of an adhesion agent every time the sprayer is filled for better absorption. What is the maximum number, x, of packs that Darius can add to the sprayer?

Tarsha is having a bake sale to raise some extra spending money for an upcoming field trip with her class. She spent $26.20 on supplies to make cookies, brownies, and cupcakes. She has already had $12.25 in sales and packaged the rest of the items at $3.25 for two cookies, one brownie, and one cupcake. What is the least number of packages, x, that Tarsha needs to sell to have some extra spending money for the field trip?

Marcie is trying to raise $26.20 to buy her dad his favorite team's cap for his birthday. She already has $12.25 saved up and her mother told her that she would give her $3.25 every time she vacuums the entire house. What is the minimum number of times, x, that Marcie needs to vacuum the house to be able to afford the cap?

Coleen is participating in the River Run Marathon that is 26.2 miles. After she completed the first 12.25 miles, they announced that participants only have 3.25 hours remaining to cross the finish line to receive a River Run Medallion. What is the slowest speed, x, in miles per hour that she must run to receive a medallion?

Reina has a mini cake pan that will hold 26.2 ounces of batter. She has found a cake mix that comes in liquid form and needs to be mixed with eggs. The more eggs used, the fluffier the cake will be. The cake mix itself is 12.25 ounces and each egg is 3.25 ounces. What values of x represent the number of eggs Reina can use and not overflow the cake pan?

 

Answer 1

These problems are not just simply 'plug and play', but you have to be able to sort out the unnecessary information while solving the problem because it can cause confusion.

 

In the first question (weed killer), there does not seem to be enough information to calculate the answer. The missing piece of information is 'how much water needs to be mixed with the herbicide'. The problem states that the herbicide is mixed with water, but not how much water per pkg, or cup.

Also, it is necessary to know if you are allowed to put just part of a packet in, or if they want you to use the whole thing.

If you are only allowed to put in whole packets, you can only use (at most) one packet because the tank only holds 26.2 cups of fluid.

One packet of herbicide contains 12.25 cups, and you have to add 3.25 cups of adhesion fluid for each herbicide packet for a total of 12.25 + 3.25 = 15.5 cups of fluid.

This doesn't account for adding any water, so you can put in one packet (12.25 cups) of herbicide PROVIDED that you don't have to add more than 26.2 - 15.5 = 10.7 cups of water per packet. If you have to add more water than that, you can't even put in a single whole packet of herbicide, but if you put in 2, the tank isn't big enough to hold them (15.5 cups x 2 = 31.0 cups, and that's WITHOUT water!)

The algebraic solution is: total tank capacity / total cups of herbicide per packet + adhesive agent

26.2 / (12.25 + 3.25) = x         all operations in parenthesis are done before other operations

26.2 / 15.5 = x

Carry out the division and you will get 1 plus some fraction of a packet.

 

 

In the second problem (bake sale), the useless information is how Tarsha is packaging her products since they don't tell you how many of each product she had initially.

 

They basically want to know when she will 'break even' on her sales (when her sales will cover her expenses).

She has spent $26.20 on supplies, and has already sold $12.25 worth of product.

Solution:

1) Subtract what she sold from what she spent to get how much more she needs to cover her costs.

2) Take that amount and divide it by how much she will make per packet and that will tell you how many packets she will need to sell. (Write the equation using 'x' to represent the number of packets because that is what you are trying to figure out)

algebraicaly

1) amount Tarsha has spent - amount she has already sold = amount she needs to break even

2) amount she needs to break even / amount she gets per packet = x (number of packets she needs to sell)

 

You will get a whole number and a fraction as an answer, but she cannot break the packet up so you have to round the value UP to the next whole number.

 

The thrid problem with Marcie is very much like the one with Tarsha. Apply the same steps.

 

The fourth problem starts off the same way as the others. You have a given amount (volume of a tank, money invested, money needed for a purchase, etc.) minus an amount that gets you part of the way there. You have to subtract the amount you have from the first amount to find out what's left. You take that and divide it by an amount per unit (cups, dollars per package, etc.) and that gives you the answer.

In this case the amount you start with is the amount of miles in the marathon, 26.2 miles. Coleen has already run 12.25 miles. Find the difference and divide by the time she has left; this will give you how many miles/hour she has to run to finish in the time she has left.

Solution:

1) miles in marathon - miles Coleen has run = miles left to go

2) miles left to go / time left in the race = x (how fast Coleen has to run to finish the race in the time left)

 

The last problem is similar to the first, but they have given you all the information you need to solve the problem. Instead of a spray tank, they give you a cake pan. They tell you its how much it can hold (26.2 ounces - the same as the tank except the tank uses cups instead of ounces). They tell you how many ounces the cake mix has, and that it is in a liquid form (you don't have to add any other liquid), and they tell you how many ounces each egg takes up (3.25 ounces). All of these measurements are the same as the first question, except they are using ounces here instead of cups!

Solution:

1) take the total capacity of the pan (26.2 ounces) and subtract the volume of cake mix (12.25 ounces) to get the number of ounces you can fill with eggs

2) take the volume you can fill with eggs and divide it by the number of ounces per egg to get the maximum number of eggs you use in the mix

** This problem is different than the bake sale. In the bake sale answer you had to round UP to get how many packets had to be sold to reach a MINIMUM amount of money. In this question, you will have to round DOWN because rounding up to the next number will require you to add an amount that will OVERFLOW the pan!

Algebraically:

1) pan volume maximum capacity - cake mix capacity = volume left in pan

2) volume left in pan / volume per egg = x (number of eggs you can put in pan to reach pan capacity (round down))

 

Good luck! ;-)