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Playlist Logic and Proof 15 videos

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Conditional Statements
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If you already know what a conditional statement is, then you're smarter than you look. And boy do you look smart.

1
Formal Proofs
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Formal proofs present the oh-so-mathematical evidence in two columns: one for the statement they are claiming is true, and the other for the reason...

2
Indirect Proofs
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Indirect proofs (or proofs by contradiction) prove their conclusion by showing that the opposite cannot be true. You can use this in math and in ar...

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CAHSEE Math 6.3 Mathematical Reasoning 171 Views


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Description:

CAHSEE Math Mathematical Reasoning Drill 6, Problem 3. If, instead, 30 total milligrams were required, which of the following equations shows the correct changes to the above original question?

Language:
English Language

Transcript

00:03

Here’s an unshmoopy question you’ll find on an exam somewhere in life…

00:07

A solution containing 30% alcohol is mixed with a solution of pure alcohol to obtain

00:12

20 milligrams of a solution that is 70% alcohol.

00:17

To determine the amount of pure alcohol that was added to the solution,

00:20

the equation 0.3 times 20 minus x… plus x… equals 0.7 times 20… could be used.

00:29

If, instead, 30 total milligrams were required,

00:32

which of the following equations shows the correct changes to the above original question?

00:37

And here are the potential answers…

00:41

That’s a huge cartload of information.

00:43

Let’s use the given equation to try to make sense of all of it.

00:47

0.3 times 20 minus x… plus x… equals 0.7 times 20.

00:53

0.3 is multiplied by 20 minus x.

00:56

0.3 refers to the solution that’s 30% alcohol.

01:00

But what does 20 minus x mean?

01:02

Well, 20 must be the resulting amount of solution, and x is the amount of pure alcohol that is added.

01:09

The solution is made up of an unknown amount of 30% alcohol and x amount of 100% alcohol.

01:17

20 minus x refers to the amount of 30% alcohol that’s used in the solution…

01:21

…because x plus the amount of 30 percent alcohol used should add up to 20 milligrams.

01:27

x plus 20 minus x equals 20. The result, 0.7 times 20, is the ending percentage

01:34

of alcohol multiplied the amount of solution or 20 milligrams.

01:41

So, now that we know what the equation means…

01:44

…what will happen to the equation if we suddenly decided that we want

01:47

30 milligrams of a 70% solution? Well, the end result will be 30 milligrams

01:52

of 70% solution instead of 20 milligrams of that 70% solution.

01:56

The new sum is 0.7 times 30, which means we can eliminate option B, which does not change

02:03

the ending total. If we want to end with 30 milligrams of solution,

02:07

then we also have to put in 30 milligrams of pure and 30% alcohol.

02:13

If x still represents the amount of pure alcohol we put in,

02:16

then x plus a mystery amount of 30% alcohol equals 30.

02:21

x plus 30 minus x equals 30.

02:23

0.3 times 20 minus x is replaced with 0.3 times 30 minus x.

02:29

That eliminates options A and D from the list, and since we already eliminated B…

02:34

…our answer is C.

02:35

As in, “Confusing.”

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