MHB How many bills of each type are there in a wallet with $\$460$?

  • Thread starter Thread starter bergausstein
  • Start date Start date
  • Tags Tags
    Money
bergausstein
Messages
191
Reaction score
0
A wallet has $\$460$, in $\$5$, $\$10$, and $\$20$ bills. the number of $\$5$ bills exceeds twice the number of $\$10$ bills by $4$, while the number of $\$20$ bills is 6 fewer than the number of $\$10$ bills. how many bills of each type are there?

i solved this problem in two ways by 1st choosing the unknown represent the number of $\$10$ bills and 2nd by choosing the unknown represent the number of $\$20$ bills.

and i got these answers from my two methods, $14$ $\$10$ bills, $8$ $\$20$ bills and $32$ $\$5$ bills.

on my third method i chose the unknown represent the number of $\$5$ bills.
and here how it goes,

let $x=$ number of $\$5$ bills,
$\frac{x}{2}-4=$ number of $\$10$ bills
$\frac{x}{2}-4-6=$ number of $\$20$ bills

$5x+10\left(\frac{x}{2}-4\right)+20\left(\frac{x}{2}-10\right)=460$
$5x+5x-40+10x-200=460$
$20x-240=460$
$20x=700$
then, $x=35$ ---> from here the number of $\$5$ bill is bigger than my previous result. can you pinpoint where is the mistake here? thanks!

p.s use only one variable.
 
Mathematics news on Phys.org
If $x$ is the number of \$5 bills then:

$$\frac{x-4}{2}$$ is the number of \$10 bills.

$$\frac{x-16}{2}$$ is the number of \$20 bills.
 
MarkFL said:
If $x$ is the number of \$5 bills then:

$$\frac{x-4}{2}$$ is the number of \$10 bills.

$$\frac{x-16}{2}$$ is the number of \$20 bills.

if i use that $x=32$ which is the number of $\$5$ bill. thanks!

if some question why did you divide $x-4$ by $2$? in my first method what the number of $\$5$ bill is $2x+4$
that's why i thought of taking the opposite operation so i came up with $\frac{x}{2}-4$.
 
Last edited:
If $x$ is the number of \$5 bills and $y$ is the number of \$10 bills, then we have:

$$x=2y+4\implies y=\frac{x-4}{2}$$
 
I think you might have solved this problem already, but just for practice can you show how to get the number of \$20 bills in terms of $x$? MarkFL showed how to get the number of \$10 bills but didn't show all the steps as to how he got $\dfrac{x-16}{2}$ for the number of \$20 bills. Can you show how to get this value? :)
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top