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Question

Paul spent 1/4 of his money on 6 notebooks and 4 thumb drives. The cost of each notebook is 2/3 of the cost of each thumb drive. How many more notebooks can he buy with 1/2 of his remaining money?

Answer 1

[NOTE 1: My tutoring is highly individualized. My multi-strategy problem solving approach leads to best results with fine-tuning to individual student’s weaknesses and strengths.]

[NOTE 2: Among many other things I teach, is using observation-reading-based shortcuts. This word problem calls for that, especially so, as we are talking about taking the SAT Test. Time-management skills are paramount in this timed test. Developed skills to spot a shortcut can be, and will be, the deciding factor on the Test taking day. Compare the following to other solutions.]

A. Tutor's Solution when working with students

Observation 1:

Given: Paul spent money on 6 notebooks and 4 thumb drives. AND

Given: Each notebook costs 2/3 of the cost of each thumb drive.

That means, he spent equal amounts (!) on notebooks and drives, as 6 * 2/3 = 4.

Then, he spent same portion of the money on each type, notebooks and drives, which is (1/4) / (2) = 1/8.

Given: A purchase considered with 1/2 of remaining, after spending 1/4, money.

Then, considered for spending is (1/2) * (1 - 1/4) = (1/2) * (3/4) = 3/8.

Observation 2:

Paul spent 1/8 of money on 6 notebooks. AND

He wants to spend 3/8 of money more on notebooks. Which is three times 1/8.

Then, he can buy 6 * 3 = 18 notebooks is our answer.

B. Student's Work on a Test when show-your-work is NOT required.

[Mental reasoning is similar to A. Have on scrap paper similar to the following (or less):]

6 nb each 2/3 of each of 4 td

6 * 2/3 = 4 That means - same amount for nb and td.

So, each, nb and td, is half of 1/4 => 1/8

1/2 of remaining is (1/2) * (1 - 1/4) = (1/2) * (3/4) = 3/8.

spent: 1/8 on 6 notebooks. AND

to spend: 3/8 , which is three times 1/8.

can buy 6 * 3 = 18 nb

[Mark option with 18 OR

Write 18 , if just space provided]

C. Student's Work on a Test when show-your-work/reasoning is REQUIRED.

[Recommended to have in the space provided:]

Given: Paul spent money on 6 notebooks and 4 thumb drives. AND

Given: Each notebook costs 2/3 of the cost of each thumb drive.

That means, he spent equal amounts on notebooks and drives, as 6 * 2/3 = 4.

Then, he spent same portion of the money on each type, notebooks and drives, which is (1/4) / (2) = 1/8.

Given: A purchase considered with 1/2 of remaining, after spending 1/4, money.

Then, considered for spending is (1/2) * (1 - 1/4) = (1/2) * (3/4) = 3/8.

Paul spent 1/8 of money on 6 notebooks.

He wants to spend 3/8 of money to buy more notebooks. Which is three times 1/8.

Then, he can buy 6 * 3 = 18 notebooks.

Answer: He can buy 18 more notebooks with 1/2 of the his remaining money.

Answer 2

If this is an SAT question, the previous answers, although very descriptive, involve too much pencil time.

Think!

1/4 spent

3/4 remain

half of that is 3/8

3/8 = 1/4 plus one-half of 1/4

He can buy (6n + 4t) + (3n + 2t) = 9n + 6t

6t = (6)(3/2)n

9n + 9n = 18n

Answer 3

Answer 4

Set the cost of one thumb drive be t dollars.

Since each notebook costs 2/3 as much as a thumb drive, one notebook costs (2/3)t.

The total amount Paul spent is:

6 notebooks + 4 thumb drives

= 6(2/3)t + 4t

= 4t + 4t

= 8t

This represents 1/4 of all his money.

Therefore, his total amount of money was:

4 × 8t = 32t

After the purchase, he had:

32t − 8t = 24t remaining.

Half of his remaining money is:

24t ÷ 2 = 12t

Since each notebook costs (2/3)t, the number of additional notebooks he can buy is:

12t ÷ (2/3)t

= 12 × 3/2

= 18

Answer: 18 notebooks.

Answer 5

We can start by turning sentences into equations.

"Paul spent 1/4 of his money on 6 notebooks and 4 thumb drives" can be written as

1/4 M = 6 N + 4 T

"The cost of each notebook is 2/3 of the cost of each thumb drive" can be written as

N = 2/3 T

Since we want to know how many more notebooks Paul can buy with half of his remaining money, we don't need to work with our T variable (cost of thumb drives). We can solve our second equation for T and plug it into our first equation to see how M (amount of money) and N (cost of notebooks) relate to each other.

T = 3/2 N

1/4 M = 6N + 4 (3/2) N

1/4 M = 6n + 6N

1/4 M = 12N

M = 48N

So if Paul spent all his money on notebooks, he could buy 48 of them. But he only wants to spend half of his remaining money. He already spent 1/4 of his money, so he has 3/4 remaining.

1/2 (3/4) = 3/8

So to see how many notebooks he can buy for 3/8 of his money, we can multiply both sides of our equation by 3/8.

3/8 M = 3/8 (48N)

3/8 M = 18 N

Therefore, Paul can buy 18 notebooks..

Answer 6

1/4 of his money (M) buys 6 notebooks (N) and 4 thumb drives (T)

(1/4)M = 6N + 4T

Notebook is 2/3 cost of thumb drive.

N= (2/3)T

Rearranged for substitution:

T= (3/2)N

He only wants to buy more notebooks. So get rid of thumb drives in 1st equation by substituting (3/2)N for T:

(1/4)M = 6N + 4(3/2)N

(1/4)M = 6N + 6N

(1/4)M = 12N

Only notebooks now, but wrong amount of money still.

He spent 1/4 of his money, so he has 3/4 of his money left. Question asks how many notebooks half of what's left can buy...:

(3/4)M x (1/2) = (3/8)M

...or in other words, how many notebooks 3/8 of his money can buy. So, find what you can multiply (1/4)M by to make it (3/8)M. That would be 3/2, so multiply both sides of the equation by 3/2.

(3/2)(1/4)M = (3/2)12N

(3/8)M = 18N

He can buy 18 notebooks.

Answer 7

Hello, thank you for taking the time to post your question!

When you have an equation like this you want to find a way to relate the variables to one another … so if N is notebooks, T is thumb drives, and M is the total amount of money you could set it up as

6N + 4T = 1/4M

N = 2/3T

From there then you can figure out how much of the total money one notebook costs by taking

6(2/3T) + 4T = 1/4M

8T = 1/4M

T = 1/32M

Then plugging back into the other equation

N = 2/3(1/32M) = 1/48M

So 1 notebook costs 1/48 of the total amount of money.

Finally then we know that half the remaining money would be ½(3/4) = 3/8. So you can take 3/8 divided by 1/48 to get 18 Notebooks.

I hope that helps get you moving in the right direction! Feel free to reach out if you still have questions beyond that :)

Answer 8

For these types of word problems, what I like to do is read the question first, and then go to the beginning.

We want to know how many more notebooks Paul can buy with 1/2 of his remaining money, and since he's already spent 1/4 of his money, that means we have to take half of 3/4, which is 3/8. So the question is asking, how many notebooks can Paul buy with 3/8 of his original money.

Now, let's set some variables.

M = Paul's total amount of money

T = Cost of each thumb drive

2/3T = Cost of each notebook (since the cost of each notebook is 2/3 the cost of each thumb drive)

Paul spent 1/4M on 6 notebooks and 4 thumb drives, so

6 * (2/3T) + 4 * T = 1/4M

Simplifying the left side gives us:

4T+4T=1/4M

8T=1/4M

And now, solve for M:

M=32T

Since we want to know how many notebooks Paul can buy with 3/8M, not M, we multiply 32T by 3/8 to get 12T.

Finally, to get the answer, all we need to do is divide how much money Paul wants to spend by the cost of each notebook to see how many notebooks he can buy.

12T/(2/3T) = 18

Final answer: Paul can buy 18 more notebooks with 1/2 of his remaining money.

Sorry for the lengthy response, hope this helps!