Algebra: Translating Words to Symbols w/1 Variable

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bergausstein
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1.translate to algebraic symbols using one variable.

a.) the excess of the square of a number over twice the number.
b.) the amount by which five times a certain number exceeds 40

it seems that "exceed" has a different meaning here.

2. Mary is 18, she is twice as old as Anne was, when Mary was as old as Anne is. How old is Anne?

let x = age of anne
2x = age of mary

now i can't continue...please help
 
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1.) I believe what they mean here is how much larger one quantity is compared to another. For example 150 exceeds 100 by 50, found by 150 - 100 = 50.

Can you give Problem 1 a try now?

2.) There are two unknowns here: Anne's age $A$ and a period of time $T$, some number of years ago in the past.

The first equation can be derived from the statements "Mary is 18, she is twice as old as Anne was":

$$18=2(A-T)$$

The second equation may be derived from the statement "when Mary was as old as Anne is". This means Anne's age now is what Mary's age was then...can you write the second equation?
 
bergausstein said:
1.translate to algebraic symbols using one variable.

a.) the excess of the square of a number over twice the number.
b.) the amount by which five times a certain number exceeds 40

it seems that "exceed" has a different meaning here.
"Exceed" has pretty much the standard meaning of "be larger than" in both of these. You can find out how much larger one number is than another by subtracting them.
a) Call "the number" x. Then its square is x^2. Twice the number is 2x. The excess of the square over twice the number is x^2- 2x.

2. Mary is 18, she is twice as old as Anne was, when Mary was as old as Anne is. How old is Anne?

let x = age of anne
2x = age of mary

now i can't continue...please help
Since we are asked for Anne's age, setting x equal to that is an obvious thing to do. But I found it simpler to let y be the difference between Mary's and Anne's ages. There are two different times involved here, NOW, and an earlier time, but the difference between their ages doesn't change. Since Mary's age now is 18, Anne is now 18- y. When Mary was 18- y, Anne was (18- y)- y= 18- 2y. Twice that is 2(18- 2y) and that is equal to Mary's age now, 18. So 2(18- 2y)= 18. Solve that for y.