Question

more algebra 2 word problems
heathers age is 3 years more than one half her cousin rachels age. if the difference in their ages is less than 4 years, what is the oldest that rachel can be in years?

Answer 1

Let the Heather's age be : x

And the Rachel's Age be : y.

So:

\[x = 3 + \frac{1}{2}y\]
\[2x = 3 + y\]
\[2x - y = 3\]

\[x - y < 4\]

\[x - 2x + 3 < 4\]
\[-x < 1 \implies x > -1\]

Answer 2

thank you!!!

Answer 3

\[3 + \frac{1}{2}y - y < 4 \implies 3 - \frac{1}{2}y<4 \implies 6-y< 8 \implies y> -2\]

Answer 4

Why thank you???

Answer 5

you helped a lot

Answer 6

I am solving it wrong..

Ha ha ha..

Age cannot be negative..

Answer 7

the question might be incomplete, but I will begin with
\[r-h<4\] or \[h-r<4\]
where
\[h=3+\frac{ 1 }{ 2 }r\]
this simply means that Rachel (r) is older than Heather (h)
\[r>h\]
by
\[r=2h-3\]
I will then pick the first inequality, r - h < 4

\[r-(3+\frac{ 1 }{ 2 }r)<4\]

Answer 8

How r is older ???

Answer 9

eathers age is 3 years more than one half her cousin rachels age

Answer 10

So heather is older no ??

Answer 11

h=\[\frac{ 1 }{ 2 }r+3\]

Answer 12

Put r = 2 there...

Answer 13

not twice, one-half

Answer 14

For r = 2, h = 4 implies h is older..

Answer 15

I was thinking that it should be h=2r+3, but that is incorrect if you put it in words.

Answer 16

Yes this is incorrect..

Answer 17

let me drink coffee and then give me 5 mins. I just woke up...

Answer 18

\[h = 3 + \frac{1}{2}r\]

This is according to question and h is older than r..

Answer 19

haha , @waterineyes , in your equations you were assuming heather is older but rachael is always older
\[x = 3 + \frac{1}{2}y\]
\[y -x < 4\]
by substitution
\[y-(3+\frac{1}{2}y) < 4\]
\[\frac{1}{2}y < 7\]
\[y < 14\]
Thus oldest rachael can be is 14

Answer 20

Their difference is less than 4 implies it can be 3, 2, 1 or it can be zero also..

x > -1, x can be 1, 2, 3, 4 etc

y > -2, y can be 1, 2, 3, 4 etc

But there is one condition also that x = 3 + 1/2y..

Answer 21

Put y = 2 there..

Answer 22

sorry ...not always but we are looking at maximizing y

Answer 23

Yeah this can be possibility @dumbcow

Thanks for correcting me..

Answer 24

There is something wrong with the question too..

The difference will be less than 4 and x = 3 + y/2

For y = 2 we get x = 4

Also x - y < 4

Both the condition satisfy..

Answer 25

let us break the words down:

Heather's age is 3 years more (+3) than one-half (1/2) of Rachel's age.

Then think if this makes sense: Rachel is 3 years younger (-3) twice Heather's age 2h.

Answer 26

r=2h-3

Answer 27

\[x = 3 + \frac{1}{2}y \implies y = 2x - 3\]

Rachel is 3 years less than twice the Heather's age..

Answer 28

this is why I picked r and h, instead of x and y for simplicity

Answer 29

My issue was not that..

I am just saying that Rachel will be younger than heather..

And yes you can take h and r for simplicity..

Answer 30

How can a person at the age of 2 younger than heather by 2 years be older ?? @dumbcow

Answer 31

Heather's age is 3 years more (+3) than one-half (1/2) of Rachel's age.

Heather's age is best understood by looking that it is first half of Rachel's age. And then add 3 years.

Answer 32

Yes but for r = 2 all situations fail..

Answer 33

Are you really typing @panlac01

Answer 34

\[r-(3+\frac{ 1 }{2 }r)<4\]

\[r-\frac{ 1 }{ 2 }r<7\]
\[\frac{ 1 }{ 2 }r <7\]
\[r<14\]

Answer 35

So, maximum \(r\) will be 13 years..

Answer 36

No it will be 12 years..

Answer 37

13 years and 11 months and 30 days

Answer 38

Ha ha ha...
For 13 years, you will get x as not an natural number..

I think r will be 12 years..

Answer 39

but that's how math works. it doesn't care when a person is half. we assume that it is 1 because there can't be a half a person.

Answer 40

It is basically when we assume x and y we assume that these are ages so ages can have only natural values..

For r = 13 you are getting h not a natural number so r = 13 is not possible..

Answer 41

it is true that it is not a natural number, but we can say that Heather is 9 years old and 6 months at minimum

Answer 42

and to go back to the question, Rachel is less than 14 years of age. Meaning, she is 13 years old, 11 months and 30 days old at most. If you round this of, you will get 14.

So I think it is safe to say that Rachel is 13-years-old and you can never be wrong for making that as a final answer.