Algebra - very basic

term

either a single number or variable, or the product of several numbers and/or variables separated from another term by a + or - sign in an overall expression

Algebraic product =

products which contain variables, such as 6c or 4x²y [40]

If I have 3 chocolate donuts and 9 vanilla donuts, what fraction of the donuts are chocolate?

1/4 => You get this answer by dividing the number of chocolate donuts (3) by the TOTAL number of donuts (3 + 9 = 12). 3 divided by 12 = 3/12 = 1/4. You get 1/4 by dividing both top and bottom by 3.

Isabella bought exactly twice as many green sweaters as pink sweaters. What could be the total number of sweaters she bought? a.) 10, b.) 13, c.) 16, d.) 18, e.) 25.

12 (D.) This is actually a much easier question than it seems. Let's say that the number of pink sweaters you buy is designated by the letter p. Therefore, the number of green sweaters must be 2 x p (twice as many). So the TOTAL NUMBER of sweaters must be p + 2p = 3p. Therefore, since you can't have fractions of sweaters, the correct answer must be divisible by 3. 18 is the only choice that falls in this category.

If 1/4 N = 12 then 1/2 N = ?

24. The easiest way is to solve for N, then plug N into the second equation (1/2 N). So if 1/4 N = 12, you try to get everything except N on the right side of the equal sign. To do this you have to get rid of the ugly fraction. How do you do this? To get rid of 1/4, multiply times 4, to make it 1 x N and remember you don't write the 1 x part so just end up with N. Problem: you can't just multiply times one side of the equation - you have to do the same thing to the other side to make them equal. So N = 12 x 4, or N = 48. If N = 48 then 1/2 N = 1/2 x 48 = 24. If you're really stumped, try to think it through: Whatever N is (unless it's a negative number), 1/2 N must be more than 1/4 N, right? so since 1/4 N is 12, any answer less than or equal to 12 must be wrong. Eliminate 3 and 6 in this case. Now, if you're really smart, you can figure out that 1/2 of something is twice as much as 1/4 of something, for the same reason that a half dollar is twice as much as a quarter. So whatever N is, 1/2 N has to be twice as much as 1/4 N. If 1/4 N is 12, then 12 x 2 = 24, and this must the answer to 1/2 N. (Note that we have not solved for N in this case, saving ourself some time, but if this gets you confused, don't do it.)

9 + 6 x (8 - 5)

27 = 9 + 6 x 3 = 9 + 18 // you have to do parentheses first

What is 2(x+3)?

2x + 6 => you must multiply 2 times x plus 2 times 3

What is 5x - x?

4x since 5x - x is really 5x + (-1)x = (5-1)x = 4x

Srijana has sold 23 more boxes of girl scout cookies than Isabella. If there are 3 weeks remaining before they have to turn in their orders, how many more boxes per week must Isabella sell to beat Srijana?

8 boxes per week. This is a little tricky, but think of it this way. Isabella is behind by 23 boxes. She has 3 weeks to close the gap. So 23 boxes divided by 3 weeks is the number Isabella must sell just to tie. 23 divided by 3 works out to 7 and 2/3. But remember, this would only tie, and the question asks how much must Isabella sell to win. Also, boxes can't be divided up, so 8 (the next whole number up from 7 and 2/3) is the right answer. In other words, if Isabella sells 8 more boxes on average per week more than Srijana over the next 3 weeks, she will sell 3 x 8 or 24 boxes. Hint: if you're really pressed for time, use the "brute force" method. Guess an answer, see if it works, then go higher or lower. For example, you might have guessed 2 boxes per week, but 2 x 3 = 6 which is less than 23, so that would not close the gap. The answer must be higher than 2 boxes per week. how about 4? Well, 4 x 3 = 12 which is still less than 23. Hmmm. How about 8? 8 x 3 is 24 and - aha - that's the right answer.

Betsy rakes leaves from 9:30 am to 11 am and finished 1/4 of the yard. She wants to finish raking by 6:30 pm. If she plans to rake at the same rate, what is the latest time she can start raking the leaves again?

This is a very hard problem and requires several steps, but each of these steps is relatively simple, so if you break it down, you can solve it. First, we must calculate the time it took her to do the work she did already. If you start at 9:30 and end at 11 am, that's 1.5 hours. She did only 1/4 of the yard in this time, though, so we know she has 3/4 of her work ahead of her. So, if she could do 1/4 in 1.5 hours, how many hours would it take her to do 3/4? Since 3/4 is 3 times 1/4, all we have to do is multiply 1.5 hours time 3 to get 4.5 hours. But we're not done. They asked us the latest time she could start to be done by 6:30 pm? So we have to subtract 4.5 hours from 6:30 pm. 6:30 - 4.5 hours is 2:00 pm., choice C. That is a hard one!

If 5 x (P + Q) = 30 and P is greater than zero, then Q could NOT be: a.) 7, b.) 4, c.) 3 1/2, d.) 0, e.) -2

a.) 7. This looks like a much harder problem than it is. All you have to do is to substitute the numbers for P one at a time to see what they're getting at. Start with the biggest and smallest number. If you were lucky enough to start with seven, you would realize that even if Q were zero (the smallest number that Q could be), 5 x (7 + 0) = 30, but 5 x 7 = 35 which is greater than 35. This tells you that P could not be seven - By the way, in a problem like this once you find any number that P could not be, you're done. Unless you made a mistake, you do not need to waste time on the other possible answers.

a(b + c) =

ab + ac => Distributive law [32]

(a + b)(c + d) =

ac + ad + bc + bd => FOIL: First, Outer, Inner, Last [33]

(a + b)² =

a² + 2ab + b²

(a - b)² =

a² - 2ab + b²

Translate into an equation: "The sum of 2 consecutive even integers is 34"

x + (x + 2) = 34 = the equation [84]

John has x dollars. Ann has $5 more than John. If Ann gives John $10, then, in terms of x, how many dollars will Ann have?

x + 5. It might help to make a table here. We start with John having x dollars. We don't know what x is and don't need to know. We know that Ann starts with $5 more than John, however much that is, so we can express this as x + 5 dollars. Now, Ann gives John $10. What does this do to our table. Well, Ann loses (minus) 10 and John gains (+) 10. So we add these to what they started with. John started with x and now has $10 more so whatever x is, he now has x plus 10. Ann started with x + 5 dollars and gave away $10 so she must now have x + 5 - 10 or x -5 dollars. This is the correct answer, C.

Solve x - 1 = 2 (x + 3)

x - 1 = 2x + 6; -x - 1 = 6; -x = 7; x = -7

if -x = 7, x = ?

x = -7 => if -x = y then x = -y

Solve 3 - 2x < 7

x >= -2 [90]