Nickels and dimes system of equations

Question

John Cena has 20 coins consisting of nickels and dimes. If his nickels were dimes and his dimes were nickels he would have 70 cents more. How much are his coins worth?

so i did $$x+y=20$$ $$2x+y=70+x+y$$ $$x=70$$ $$y=-50 $$

but this obviously isnt the answer and i don't know what to do i need help

Answer 1

Hint: Use the equations $$n+d=20$$ and $$n+0.5d=0.5n+d+7$$ If you need further elaboration I will explain.

Answer 2

So, you were right in how you started. I'll repeat what you did so far.

We want to figure out how many dimes and nickels we have. Let us represent the number of dimes by $x$ and the number of nickels by $y$.

Since we have $20$ coins in total, we know $x + y=20$.

How much money is this? Well, a nickel is worth $.05$ ($5$ cents) and a dime is worth $.1$ ($10$ cents). So, with $x + y$ amount of coins ($x$ being dimes and $y$ being nickels), we have $.1x + .05y$ amount of money.

Now we know that if we had as many nickels as dimes, and as many dimes as nickels, we would have $.7$ more ($70$ cents more). That means our original amount of $.1x + .05y$ plus $.7$ should equal the amount we have if the amount of nickels and dimes were switched. So, we have the equation: $$.1x + .05y + .7 = .1y + .05x $$

This equation can be simplified to be $.05x - .05y = -.7$.

Now, we have two equations and two unknowns:

$ x + y = 20$

$.05x - .05 y = -.7$

Multiplying the first equation by $.05$ allows us to rewrite the equations as:

$.05x + .05y = 20*.05 = 1$

$.05x - .05y = -.7$

Now, adding the two equations above gives:

$.1x = .3$

And solving for $x$ gives $x = .3/.1 = 3$.

Since $x$ was the number of dimes, it means we have $3$ dimes. Since we started with $20$ coins in total, it means we have $20 - 3 = 17$ nickels.

From here, you can figure out how much the money is worth by computing $17*.05 + 3*.1$.

Answer 3

Solve the following system of $2$ equations in $2$ variables:

• $5x+10(20-x)=y$
• $10x+5(20-x)=y+70$

Which leads to:

• $-5x+200=y$
• $5x+100=y+70$

And then to:

• $300=2y+70$

And then to:

• $115=y$ we have essentially answered the question at this point

And then to:

• $-5x+200=115$

And then to:

• $x=17$

Which means that his coins are worth $5\cdot17+10\cdot(20-17)=115$ cents.

Answer 4

If you don't like working with non-integers, you could "change units" and think of American currency in terms of nickels. However, until you're comfortable working with word problems (in particular being able to follow and replicate the "standard" approaches in the other answers), I suggest trying to follow the other answers first before trying to follow this one.

Viewing currency in units of nickels, we can say that if his nickels and dimes were interchanged, then the amount of additional money he would have is equivalent to 14 nickels. If $x$ and $y$ are the number of nickels and dimes he has, then $x+y=20$ (as you correctly stated), the total worth of his coins is $x+2y$ (since a nickel is worth $1$ nickel and a dime is worth $2$ nickels), while the total worth of his coins were they reversed would be $2x+y$. The last two bits of information combine to give $$(2x+y) - (x+2y) = 14$$ which simplifies to $$x - y = 14.$$ The system of equations $$ x+y = 20 \\ x-y = 14 $$ yields $x = 17$ and $y = 3$, from which you can calculate that his total worth (in nickels!) is $$ x + 2y = 17 + 2(3) = 23. $$ So you have the amount of money he owns, in nickels. Now convert that to dollars or cents by using the appropriate conversion.