SOLUTION: in a block of flats there are 24 units of 3 types: the luxury unit, the superior unit and deluxe unit. the luxury can accommodate 8 people, the superior unit can accommodate 7 peop

Question 1179804: in a block of flats there are 24 units of 3 types: the luxury unit, the superior unit and deluxe unit. the luxury can accommodate 8 people, the superior unit can accommodate 7 people and deluxe can accommodate 5 people. given that the total number of people living in this block is 160. How many of each type are there?
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!
"Can accommodate" does not ensure full capacity.
Answer by ikleyn(52798)   (Show Source): You can put this solution on YOUR website!
.
in a block of flats there are 24 units of 3 types: the luxury unit, the superior unit and deluxe unit.
The luxury can accommodate 8 people, the superior unit can accommodate 7 people and deluxe can accommodate 5 people.
Given that the total number of people living in this block is 160, how many of each type are there?
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Let x be the number of the luxury units; y be the number of the superior units. Then the number of the deluxe units is 24 - x - y. Assuming maximum accommodation (fulfillment) of the units, we write this equation 8x + 7y + 5*(24-x-y) = 160. We simplify it 8x + 7y + 120 - 5x - 5y = 160 3x + 2y = 160 - 120 3x + 2y = 40 x = . We want have x and y as integer numbers. It gives for x and y these values (see the TABLE) T A B L E x y z luxury superior deluxe --------------------------- 12 2 10 10 5 9 8 8 8 6 11 7 4 14 6 2 17 5 0 20 4 In parallel, we fill the column for z = 24 - x - y. We do it until we have non-negative values for x, y, and z. This list in the table is the FULL SET of all possible solutions to the given problem. ANSWER

Solved.

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Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

x = # of luxury units
y = # of superior units
z = # of deluxe units

the total number of units is 24
the total number of residents is 160

This is two equations with three variables, in which the values of the variables are non-negative integers. That is a system of Diophantine equations; there are standard processes for solving them.

We first need to eliminate one variable to get a single equation with two variables. Let's eliminate z using elimination.

Next we solve this single equation for one variable and use the requirement that the solutions be non-negative integers to find the solution(s) to the problem.

We could at this point choose non-negative integers values for x and find the ones that give non-negative integer values for y. But we can make finding the solutions easier by going one step further with our algebra before we do that.

Perform the indicated division, expressing the result as a quotient and a remainder:

From this we see that y has to be an integer and 20-x is an integer, so x/2 has to be an integer. That tells us x has to be even.

Now we are ready to find the solutions to the problem.
x y=(20-x)-x/2 z=24-(x+y) ---------------------------- 0 20-0=20 24-20=4 2 18-1=17 24-19=5 4 16-2=14 24-18=6 6 14-3=11 24-17=7 8 12-4=8 24-16=8 10 10-5=5 24-15=9 12 8-6=2 24-14=10

Larger values for x would produce negative values for y, so the list of solutions is complete.

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