Question

Solve.
Savannah as a collection of quarters and dimes worth $3.70. She has a total of 19 coins. How many quarters and how many dimes does Savannah have?

Answer 1

let Q be the number of quarters, and D be the number of dimes. 25Q will give you the value of the quarters (in cents) and 10D will give the value of the dimes (also in cents). You can write two equations here: one shows that the # of dimes + # of quarters = the total number coins, and the other that the value of the quarters + the value of the dimes = 370 cents. Can you write and solve those equations?

Answer 2

Ummm, I'm trying..

Answer 3

write the equations first, I'll check them

Answer 4

like ,do you mean d+q=19 ?

Answer 5

yep, that's one of them. how about the one that figures out the value of the coins?

Answer 6

hmm , one moment

Answer 7

2d + 14q = 3.70 ? will that one work ?

Answer 8

no, where are these 2 and 14 coming from?

Answer 9

d is the number of dimes you have. q is the number of quarters. if you wanted to know the value of d dimes, what would you do?

Answer 10

Well what I did was found numbers of the coins that would add up to 3.70

Answer 11

but 2+14 doesn't equal 19, does it?

Answer 12

ohhhhhhhhhh , I wasn't thinking about the 19 at all , oh gosh I was only thinking about the total... you're right, you're right.

Answer 13

so, hold that thought, we'll come back to it!

Answer 14

our first equation was d + q = 19, representing that there are 19 coins, all either dimes or quarters, agreed?

Answer 15

agreed.

Answer 16

that isn't enough to uniquely determine the mix of coins we've got. we have two unknowns, so we need two equations. they'll actually be lines that intersect if there is a solution, and the point of intersection will be the solution.

Answer 17

for our second equation, we'll rely on the fact that the value of the dimes plus the value of the quarters adds up to 370 cents (I do it in cents so that we don't have any decimals to worry about)

Answer 18

if the value (in cents) of a stack of d dimes is 10d, what would the value (in cents) of a stack of q quarters be?

Answer 19

9

Answer 20

right ?

Answer 21

how could the value in cents of any number of quarters be 9 when quarters are worth 25 cents each? :-)

Answer 22

omg , you're making me feel slow lol . I swear I keep thinking about everything other than that...

Answer 23

if you have q quarters, that's going to be worth 25*q cents, right?

Answer 24

it's too simple, that's the problem :-)

Answer 25

so 10*d + 25*q = 370, right?

Answer 26

yes.

Answer 27

so our equations describing this system are
\[d+q=19\]\[10d+25q=370\]Do you know how to solve the system, now that we have the equations?

Answer 28

Hmm, I believe so.

Answer 29

try it, and report back

Answer 30

okay. I will.

Answer 31

okay...I'm kinda having problems on where to start..

Answer 32

No problem, I can teach you :-)

There are two straightforward methods that can be used here: substitution and elimination.

With substitution, we solve one of the equations to give us 1 variable (with a coefficient of 1) all by itself in terms of the other. We then substitute (hence the name) that into the other equation, giving us an equation in terms of just the other variable, which we solve. Having gotten the other variable's value, we plug it into the substitution equation and find our first variable.

Can you solve \[d+q=19\] for \(d\) in terms of \(q\) (or vice versa, it doesn't matter)

Answer 33

I wanna say...subtract something from both sides but I don't wanna be wrong..

Answer 34

That's a good choice. Anything in particular you'd like to subtract? :-)

Answer 35

q ?

Answer 36

that works. that gives us \[d = 19-q\] right?

Answer 37

yess.

Answer 38

Okay, now take the other equation:\[10d + 25q = 370\]and replace \(d\) everywhere you see it with \((19-q)\)

Answer 39

(that's our substitution step)

Answer 40

okay after substituting.. I got 10(19-q)+25q=3.70 then....
190-10q+25q=3.70 , right so far ?

Answer 41

that looks good to me

Answer 42

oops, 370, not 3.70

Answer 43

we are working in cents, not dollars. otherwise, our second equation would have been \[0.10d + 0.25q = 3.70\]

Answer 44

and the first thing I would suggest in that case would be to multiply everything by 100 to make it \[10d + 25q = 370\] :-)

Answer 45

okay okay. so after making that change.. I subtracted 10q from both sides....then I got 190+35q=370 , then subtract 190 from both sides and then that left me with 35q=180 ? right ?

Answer 46

my cat is attempting to contribute some advice by walking on the keyboard :-)

Answer 47

haha , awwww.

Answer 48

since when is -10q + 25q = 35q?

Answer 49

(that's the cat talking, not me! ;-)

Answer 50

oh nooo , I added a positive 10...

Answer 51

okay, so if you do the addition correctly, you get \[190+15q=370\]\[15q=180\]right?

Answer 52

right !

Answer 53

Even the cat knows that if \(15q=180\) then \(q=12\). So now we know that there are 12 quarters in the mix. How many dimes? Well, we take our substitution equation: \[d = 19-q\]and plug in \(q = 12\) to get\[d = 19-12 = 7\]7 dimes and 12 quarters.

Now, very important! We take our proposed solution and we plug it into all of the original equations and make sure that true statements are obtained by doing so! It isn't enough to check only one equation; you have to check them all because it is possible to get a set of numbers which will satisfy some but not all of the equations.

First equation: d+q=19
7+12=19 good so far
Second equation: 10d + 25q = 370
10*7 + 25*12 = 370
70+300=370
370=370
okay, that checks out!

Now let's talk elimination!

Answer 54

Elimination is where we add or subtract the equations to form new equations with some of the variables eliminated (again, another imaginative name, eh?)

\[d + q = 19\]\[10d + 25q = 370\]
We can do just about anything to an equation so long as we do it to both sides. For example we can multiply an equation by a number, or add a number to both sides, and the equation is still true. 2+2 = 4, and 3*2 + 3*2 = 3*4

What if we multiply the first equation by -10? We get\[-10*d+(-10)*q = -10*19\]\[-10d-10q=-190\]Now let's stack that up with the other equation and add them together, just like adding columns of numbers:
\[-10d-10d = -190\]\[10d+25q=370\]----------------

What do you get if you add down the columns?

Answer 55

15q = 180 , right?

Answer 56

exactly! and we already know what to do from here, right?

Answer 57

yes! Thank you OH SO MUCH! Thnak your cat for me too. : )

Answer 58

elimination is easy when you can spot the combination of numbers to multiply by to get one of the columns to cancel out. sometimes you have to think about it a little, but if all else fails, multiply the 1st by the coefficient from the 2nd, and vice versa. For example:

\[3x+2y=0\]
\[2x+3y=4\]
I would multiply the 1st by -2 and the second by 3 giving\[-6x-4y=0\]\[6x+9y=12\]and there the x will be eliminated.

Answer 59

now, remember I said to hold that thought some time back?

Answer 60

yes

Answer 61

Okay, here's sort a seat of the pants way to solve this. to refresh, we've got 19 coins, dimes and quarters, totaling $3.70. how many of each?

let's assume that they are all dimes. How much would 19 dimes be worth?

Answer 62

$1.90, right? but that's too small, we need $3.70. We're off by $3.70-$1.90 = $1.80. What if we exchange some of those dimes for quarters? Each dime turned into a quarter gives us another $0.25-$0.10 = $0.15 on the total, right?

Answer 63

yes.

Answer 64

So, if we convert 10 dimes to quarters, we'll have $1.90 + 10*$0.15 = $1.90 + $1.50 = $3.40. Close, but we're still 30 cents short. Change 2 more coins and we're done. 19-10-2 = 7 dimes

Answer 65

We could also have just said $1.80 / $0.15 = 12 and gotten there in a single step.

Answer 66

sometimes guessing an approximate answer and tweaking it until it fits exactly is as fast or faster than solving it exactly in the first place. depends on the size of the numbers, how quick you are at doing algebra, etc.

Answer 67

thanks so much for the help and the advice. I really do appreciate it.

Answer 68

you're welcome! now you should be able to tackle any coin problem you see :-)

Answer 69

remember, work in cents or in dollars, but don't mix them!

Answer 70

Alrighty. I will!

Answer 71

just like you can't measure in a combination of inches and centimeters...

Answer 72

right right.