Explanation:
by recognizing the numbers of the description and find the same numbers in the equations and terms.
where do you find 12 in the problem description ? ah, it is the money she gets for a recycled bottle.
where do you find 10 in the problem description ? it is the money she gets for a recycled can.
and, surprise, surprise, "4b + 1" is what Solomon found to be his number of cans.
and 218 ? come on ! how many cents are in $2.18 ? well, how many cents are in $1 ? 100. so, in $2.18 there are 218 cents !
because all the other money numbers are in cents, and we always want to compare apples with apples, the total sum of earned money is best represented in cents too.
so,
12b = 12 × number of bottles recycled = the total money earned by recycling bottles.
10(4b+1) = 10 × number of cans recycled = the total money earned by recycling cans.
218 is again the total amount of money earned by recycling bottles and cans.
the parts of Marcus' equations represent the exact same things. Marcus only introduced an intermediate step with a dietary variable (c) that Solomon simply covered implicitly.
Marcus defined the variable c to represent the number of cans (and assigned it the same term "4b+1" that Solomon just called "the number of cans").
since c is exactly representing 4b+1, Marcus' equation is the same as Solomon's. if we replace "c" by its equal representation "4b+1" we get exactly Solomon's equation.
so, yes, they represent the same problem.
12b + 10(4b + 1) = 218
12b + 40b + 10 = 218
52b = 208
b = 4
she recycled 4 bottles.
the number of recycled cans is
4b + 1 = 4×4 + 1 = 16 + 1 = 17
zu, she recycled 17 cans.