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I'm not familiar with the 5-step process, but I
can show you how I'd do it
Let = the width of the border
The area of the garden is yd2
The dimensions of the garden + border are:
by
The area of the garden + border is

The area of the garden is yd2
The area of the border is yd2




Solve using quadratic formula







(there's another solution, but it's negative)

The border around the garden is 2 yds wide
check:




OK

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2nd grade? I was taking remedial sandbox and advanced crayons.
This is a problem where she has to come to some logical
conclusions and put them together to get what they're
looking for
They don't say anything about it, but any number of pennies, nickels,
and dimes may be used
I focused on "Jake has more than $1"
(case #1) Jake could have 3 quarters and 3 dimes, giving him $1.05
(case #2) or, he could have 4 quarters, and it doesn't matter what the other 2 coins are
(case #3) 0r, he could have 5 quarters, and it doesn't matter what the other coin is
Now I look at "Jake's coins are worth 3 times Ellie's coins"
So, Jakes coins must be evenly divisible by 3
for case #1: 1.05/3 = .35
So, if Ellie has 12 coins, and I know 2 of them are quarters, she's already got
$.50, so case #1 doesn't work
for case #2: Jake could have:
$1.20 (4 quarters and 2 dimes)
$1.10 (4 quarters and 2 nickels)
$1.15 (4 quarters , a nickel and a dime)
$1.11 (4 quarters, a dime and a penny)
$1.06 (4 quarters, a nickel, and a penny)
$1.02 (4 quarters and 2 pennies)
The right one has to be evenly divisible by 3
$1.20/3 = .40
$1.11/3 = .37
$1.02/3 = .34
Now Ellie has 1 quarter, so her 11 remaining coins must add up to
.40 - .25 = .15
or
.37 - .25 = .12
or
.34 - .25 = .09
There's no way to get the last 2 with 11 coins, but
.15 = 10 pennies and a nickel
So, she ends up with 1 quarter, 10 pennies, and 1 nickel
when Jake has 4 quarters and 2 dimes
Now I just have to deal with case #3
Jake has either:
5 quarters and a dime = $1.35
5 quarters and a nickel = $1.30
5 quarters and a penny = $1.26
$1.35/3 = .45
$1.26/3 = .42
Nellie has 12 coins to make $.45
That would be 5 pennies, 6 nickels, and 1 dime
I can't seem to make $.42 with 12 coin, but no quarters
So, what I finally come up with is:
--------------------------------
When Jake has 4 quarters and 2 dimes,
Nellie has 1 quarter 1 nickel, 10 pennies
---------------------------------
When Jake has 5 quarters and 1 dime,
Nellie has 1 dime, 6 nickels, and 5 pennies
---------------------------------
Whew! What grade is she in?
This was hard for me. I'm still not sure I've
got all the answers and I've got 2 years of college

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How long does it take to go 1 mile?


hr
It will take times longer to go miles
hr

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Let = number of graphing calculators bought
= number of scientific calculators bought





20 scientific and 29 graphing were bought
check:




OK

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This says

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The formula to use when given 2 points is:

where the given points are:
(x1,y1) and (x2,y2)
It doesn't matter which is which, but I can call
(x1,y1) = (-2,0)
and
(x2,y2) = (7,4)
Putting these into the formula:


Multiply both sides by




answer

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This says 2 things:
(1)
and
(2)
Subtract from both sides of (1)
(1)
(1)
Divide both sides by and reverse < sign
(1)
or,
(1)
Now subtract from both sides of (2)

Divide both sides by and reverse sign

Or, what is the same thing:

Putting (1) and (2) together:
answer

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answer

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Multiply both sides by


take the log of both sides



You can solve with calculator

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Let = speed of helicopter with no wind
= speed of wind
given:
with the wind:

against the wind:

---------------
This is 2 equations with 2 unknowns
solve for and

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Let = toll for 1 car
Let = toll for 1 truck
given:





solve for , then

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Let = liters of 60% solution needed
Let = liters of 80% solution needed
given:
= acid in 60% solution
= acid in 80% solution
-----------------------
(1) liters


(2)
Multiply both sides of (1) by and
subtract from (2)
(2)
(1)



15 liters of 60% solution and 5 liters of 80% solution are needed

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There are 2 similar right triangles:
One is formed by her and her shadow
The other is formed by the streetlight
and (distance to her + her shadow)
I can write a ratio of the sides
of these similar triangles:
Let = the height of the streetlight


multiply both sides by


The streetlight is 22 ft high

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Let = ml of 8% solution needed
Let = ml of 3% solution needed
given:
hydrogen peroxide in 8% solution:
ml
hydrogen peroxide in 3% solution:
ml
-----------------



(1)
and
(2)
Multiply both sides of (2) by
and subtract (2) from (1)
(1)
(2)


and, since



100 ml of 8% solution and 400 ml of 3% solution are needed
check:




OK

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There are 8 liters of a mixture of 30% sprite and 70% lime juice
The liters of sprite is:

The liters of lime juice is:

Call the liters of sprite added to this to get 40% sprite




liters
4/3 liters of sprite must be added
check:
The % of lime juice is:




OK

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The 2 cars and their starting point form a right triangle
One leg is mi
Call the north leg
Then the distance between the cars is





The northbound car had gone 20 mi
check:




OK

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Plot on the vertical axis, and on
the horizontal axis
You plot it by plugging values of into the equation
and finding , like


Here's the plot:

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answer

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answer

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How long does it take to drive mi?


hr
It takes times as long to drive mi
hr

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Playing with my calculator, I found that


And I found that


so,

answer

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Let = liters of 60% solution needed
Let = liters of 80% solution needed
given:
(1) liters


(2)
Multiply both sides of (1) by and subtract (1) from (2)




and, since



15 liters of 60% solution and 5 liters of 80% solution are needed
check:
In words:
(liters of acid you end up with) / (final total liters of solution) = 65%



OK

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Let = number of nickels
Let = number of dimes
Let = number of quarters
given:
(in cents)


---------
By substitution:




I know that , since quarters must add up to less than $4.50
If then, suppose , could be
That's $2.75 plus $1.00 = $3.75. If there are twice as many dimes as
nickels, , totaling $.25
$3.75 + $.25 = $4.00
My next guess is ,,
You can go on from here, since you get the idea

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You can break it up like this:

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Let = stock's original value
= stock's value minus 20%
Let = percent that it must increase
given:


---------------





The stock must increase by 25%

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To correct your spelling:
What are the 3 consecutive integer numbers that sum 117?
Call the consecutive numbers , , and
given:






The numbers are 38, 39, and 40
check:


OK

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First find the cost of 1 box
dollars/box
Let = number of boxes
= cost of boxes

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x-intercepts occur where
The quadratic equation is

when the equation is in the form







The only x-intercept is at

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The equation is in the form
where is the slope, so

Any line parallel to this line will also have
I can write this equation as

To find , I just plug in the (x,y) = (0,2)


So, the equation is

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Let = rate of boat in still water
Let = rate of the current
Write 2 equations, one for upstream and one for downstream
given:
hrs
hrs
mi
-----------
upstream:


(1)
downstream:


(2)
-----------------
(1)
(2)
Add the equations

mi/hr
And, since
(1)

mi/hr
The rate of the boat in still water is120 mi/hr
The rate of the current is 40 mi/hr
check:
(1)


and
(2)


OK

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Use the quadratic formula










and, also



The factors are:


check:
Multiplying these, I get

Multiply both sides by


OK