Question 1186985
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Here first is a refinement, using logical analysis and simple mental arithmetic, of the informal solution shown by the other tutor...<br>
The total cost of all the items is $296, and the total cost of the shoes, at $20 per pair, is a multiple of $10.
That means the total cost of the dresses, at $24 each, must be a number with units digit 6.
That means the number of dresses must be 4, 9, 14, ...
Since 14 dresses at $24 each is more than $296, there are exactly two solutions -- with either 4 or 9 dresses.<br>
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And here is how to find the solution(s) using formal mathematics....<br>
x = # of dresses at $24 each
y = # of (pairs of) shoes at $20 each<br>
The total cost was $296:<br>
{{{24x+20y=296}}}<br>
Simplify...<br>
{{{6x+5y=74}}}<br>
Solve the equation for one variable<br>
{{{5y=-6x+74}}}<br>
{{{y=(-6x+74)/5}}}<br>
Perform the indicated division as quotient plus remainder, using integers -- instead of fractions<br>
{{{y=((-5x+70)+(-x+4))/5 = (-5x+70)/5+(-x+4)/5}}}
{{{y=(-x+14)+(-x+4)/5}}}<br>
In that form of the equation, x and y are non-negative integers, and 14 is an integer, so {{{(-x+4)/5}}} has to be an integer.<br>
Inspection shows that x must be 4, or 4 plus or minus some multiple of 5.<br>
Use that and the knowledge that x and y are both non-negative integers to find the solution(s).<br>
(1) x=4<br>
{{{y=(-4+14)+(-4+4)/5 = 10+0 = 10}}}<br>
ANSWER: x=4 dresses and y=10 pairs of shoes<br>
CHECK: 4($24)+10($20) = $96+$200 = $296<br>
(2) x=9<br>
{{{y=(-9+14)+(-9+4)/5 = 5-1=4}}}<br>
ANSWER: x=9 dresses and y=4 pairs of shoes<br>
CHECK: 9($24)+4($20) = $216+$80 = $296<br>
(3) x=14<br>
14($24) is greater than $296...<br>
There are exactly two answers:
(1) 4 dresses and 10 pairs of shoes
(2) 9 dresses and 4 pairs of shoes<br>