How many juice bottles are in each carton?

[johnnyies]

some more math :(

Homework Statement

The average of 7 consecutive numbers is 2009. What is the average of the
first three of these numbers?

Samantha went shopping. She bought the following fruit: apples, oranges
and peaches. The apples and oranges weigh 13 pounds together. The oranges
and peaches weigh 15 pounds together. The apples and peaches weigh 16
pounds together. How many pounds of fruits did Samantha buy altogether?

Andrea and Maya are helping their mother plant flowers in the garden. It
will take Andrea 3 hours to plant all the flowers working alone, and it will
take Maya 6 hours to plant all the flowers working alone. If Andrea and Maya
work together, how many hours will it take them to plant all the flowers?

Lucas is buying juice bottles for his soccer team. Juice bottles are sold by the
carton. If he buys 3 cartons of juice bottles, he will be short 3 bottles. If he
buys 4 cartons of juice bottles, he will have 3 bottles extra. How many juice
bottles are in each carton?

Homework Equations

None

The Attempt at a Solution

These are absolutely killing me. I have no idea where to begin on these. If anyone can help, or even tickle around to give me some ideas, that would help.

 

[CompuChip]

In all these questions, I suggest introducing an appropriate variable x. For example, in the first one, let x be the first of the seven numbers. Then you can write the average of 7 consecutive numbers as
$$\frac{x + (x + 1) + (x + 2) + \cdots + (x + 6)}{7}$$
From this you can easily calculate x, then find the answer.

For the second one, let a, p and o be the total weight of apples, peaches and oranges, respectively. Then you can write down three equations in the three unknowns, from the given data.

Etc.

 

[Mark44]

Lucas is buying juice bottles for his soccer team. Juice bottles are sold by the
carton. If he buys 3 cartons of juice bottles, he will be short 3 bottles. If he
buys 4 cartons of juice bottles, he will have 3 bottles extra. How many juice
bottles are in each carton?

The place to start on this problems is to define variables that represent unknown quantities, and then translate the information from prose into one or more mathematical equations.

For the problem the goal is to find the number of juice bottles in a carton.
Let n = the number of juice bottles in a carton.

We also need to know the number of players on the soccer team.
Let p = the number of players on the soccer team

Now translate the sentences in the problem to equations.
If he buys 3 cartons of juice bottles, he will be short 3 bottles.
What equation represents this sentence?

If he buys 4 cartons of juice bottles, he will have 3 bottles extra.
What equation represents this sentence?

If you correctly translate the two sentences, you will have a system of equations that you can solve for n (number of bottles per carton) and p (number of players).

 

[HallsofIvy]

Homework Statement

The average of 7 consecutive numbers is 2009. What is the average of the
first three of these numbers?

The average of any arithmetic sequence is the same as the average of the smallest and largest numbers and, of course, "consecutive numbers" is an arithmetic sequence with common difference 1. If the first number is n, then the seventh number is n+ 6. The average of those two numbers, and so the average of all 7 is (n+n+6)/2= n+ 3= 2009 (the "middle" number in the sequence, of course). That is, n= 2006 so the first three numbers are 2006, 2007, and 2008.

Samantha went shopping. She bought the following fruit: apples, oranges
and peaches. The apples and oranges weigh 13 pounds together. The oranges
and peaches weigh 15 pounds together. The apples and peaches weigh 16
pounds together. How many pounds of fruits did Samantha buy altogether?

Let A be the weight of the apples, O the weight of the oranges, and P the weight of the peaches, alll in pounds. The A+ O= 13, O+ P= 15, and A+ P= 16. Solve those three equations for A, O, and P separately. The answer is, of course, A+ P+ O.

Andrea and Maya are helping their mother plant flowers in the garden. It
will take Andrea 3 hours to plant all the flowers working alone, and it will
take Maya 6 hours to plant all the flowers working alone. If Andrea and Maya
work together, how many hours will it take them to plant all the flowers?

When two people work together, the rates of work add. Andrea is working at a rate of 1 job/3 hours= 1/3 jobs per hour. Maya is working at a rate of 1/6 job per hour. Together they work at 1/3+ 1/6 jobs per hour.

Lucas is buying juice bottles for his soccer team. Juice bottles are sold by the
carton. If he buys 3 cartons of juice bottles, he will be short 3 bottles. If he
buys 4 cartons of juice bottles, he will have 3 bottles extra. How many juice
bottles are in each carton?

Let n be the numbger of juice bottle in each carton and m be the number of juice bottles he is expected to buy. 3n= m- 3 and 4n= m+ 3. Solve those two equations for n and m. Of course, the answer to this question is m.

Homework Equations

None

The Attempt at a Solution

These are absolutely killing me. I have no idea where to begin on these. If anyone can help, or even tickle around to give me some ideas, that would help.

 

[CompuChip]

There you go, we've practically answered most of the questions for you.
Now you give it a go.