A student asked me to solve this algebraically. So, here it goes:

p pencils cost c cents at the same rate
So, the COST PER PENCIL = c/p CENTS

ASIDE: if you're not sure how to find price per pencil, you might first examine a few examples.
For example, if 3 pencils cost 6 cents, then the cost per pencil = 2 cents (notice that 6/3 = 2)
Likewise, if 4 pencils cost 20 cents, then the cost per pencil = 5 cents (notice that 20/4 = 5)
Likewise, if 7 pencils cost 63 cents, then the cost per pencil = 9 cents (notice that 63/7 = 9)
So, if p pencils cost c cents, then the cost per pencil = c/p cents

How many pencils can be bought for d dollars?
Our price per pencils is in CENTS
So, we must convert d dollars to CENTS.
Well, d dollars = 100d CENTS

ASIDE: Once again, if you're not sure how to convert d dollars to CENTS, you might first examine a few examples.
4 dollars = 400 cents, since there are 100 cents in 1 dollar (notice that 4 x 100 = 400)
Likewise, 9 dollars = 900 cents, since there are 100 cents in 1 dollar (notice that 9 x 100 = 900)
Likewise, 15 dollars = 1500 cents, since there are 100 cents in 1 dollar (notice that 15 x 100 = 1500)
Applying the same logic, we can see that d dollars = 100d CENTS

Number of items we can buy = (amount of money we have)/(price per item)
= (100d CENTS)/(c/p CENTS)
= (100d)(p/c)
= 100dp/c

Answer: E

Cheers,
Brent

Basically, we can think of this as a conversion problem where we are converting dollars to pencils.
We are given d dollars
and a ratio of p pencils/c cents

To convert the d dollars to pencils we must first convert d dollars to cents
(by multiplying by 100, b/c there are 100 cents per dollar) then we can multiply by the ratio of pencils to cents
the cents units will cancel leaving the # of pencils

d • \(\frac{p}{c}\)

d(100) • \(\frac{p}{c}\)

\(\frac{100dp}{c}\)

I show it all in detail in the attached file. Personally, I think the use of d and c for # of dollars and # of cents is confusing when writing everything out because we have to cancel units, and it becomes unclear what's being cancelled.

This is a simple "Proportions" problem.
Please ignore all the "bullshit", strategies or whatever. Do not plug in numbers. ONLY CREATE A PROPORTION and solve for X:


In order for c to be equal to d, you to have to multiply d by 100 (because you want to express it in dollars)

If you don't understand this solution, it's OK. It takes time to come up with this method.
But it's really fast when you learn how to do it. Always form a proportion on these problems.

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