Question

Josh, Ben and Rick have 70 marbles altogether. The ratio of the number marbles Josh has to the number of marbles Rick has is 1: 2. However, Ben has 15 marbles fewer than Rick. How many marbles does Ben have?

Answer 1

Answer:

Ben has 19 marbles

Step-by-step explanation:

First taking that josh and ricks ratio is 1:2, this would set up in an equation as $2j = r$ and with ben having 15 less than rick, his formula would be $r = b + 15$ as well as the total of all the boys marbles together being this $j+r+b=70$

Now we put all of these equations into a set to answer them.

$2j = r\\ r = b + 15\\j+r+b=70$

First we solve for j

$2j = r\\ j=\frac{1}{2} r$

Then substitute j

$j+r+b=70\\ \frac{1}{2}r+r+b=70\\ 2b+3r=140$

Then simplify and multiply by 2 to even out

$r = b + 15\\ -b+r=15\\2b+3r=140$

Now eliminate the b variable and combine equations

$2b+3r=140\\-2b+2r=30$

$5r=170$

Now divide and solve for r

$5r=170\\r=34$

Now we know Rick has 34 marbles of the 70 marbles, so we can substitute his number back into the original equation

$-b+34=15\\b=19$

Now we know Ben has 19 marbles of the remaining 36, and can substitute his number as well

$j=\frac{1}{2}(34)\\j=17$

And now we know how much everyone has, to check we rewrite all the equations with their respective numbers

$11+34+19=70\\2(17)=34\\34=19+15$

$70=70\\34=34\\34=34$

Which means the ordered pairing of (b, j, r) is (19, 17, 34)