You have $60 to buy fish for a 30-gallon aquarium. Each angelfish costs $12 and needs at least 6 gallons of water. Each neon tetra costs $3 and at needs least 3 gallons of water. How many of each kind of fish can you buy?

1 Answer
Feb 13, 2017

8 tetra and 3 angel fish

Explanation:

#color(blue)("Preamble")#

There is a hidden cheat imbedded in this question.

Its all down to the wording 'at least'. This indicates that you are allowed to make adjustments.

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Let Angel fish be #a#
Let Tetra fish be #t#

..................................
For given volume of water :

#a-> ul("at least")# 6 gallons
#t-> ul("at least")# 3 gallons

So for water volume equivalent #2t=a#
Max count #a->30-:6=5" Angel fish"# (with no tetra)
Max count #t->30-:3 =10" Tetra fish"# (with no angel)
............................................
For given cost:
#a->$12#
#t->$3#

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#color(blue)("Method:")#
Start at maximum Tetra gradually reducing them but being replaced by the equivalent angel fish for volume needs. Continue until the any adjustment results in the cost of $60. The cost being fixed it must be obtained.

#color(blue)("Calculations")#

#color(blue)("Try 1")#
By volume:#->t(10-1)+2a=30#
#" "9t" "color(white)(.)+2a" "=30#
By cost:#" "->9(3)color(white)(.)+2(12)=27+24=$51#

Cost difference #->60-51=$9 larr#the exact cost of 3 tetra

Adding 3 tetra would be too heavy a demand on oxygen so we have:
Cost#->$51#
Fish count#->9+2=11#
...................................................................................................................

#color(blue)("Try 2")#
By volume:#->t(10-2)+4a=30#
#" "8t" "color(white)(.)+4a" "=30#
By cost:#" "->8(3)color(white)(.)+4(12)=24+48=$72#

Cost difference #->72-60=$12 larr#the exact cost of 1 angel fish
Reduce by 1 angel fish we are increasing the associated available oxygen satisfying the 'at least' criterion. So we have:

Cost#->$60#
Fish count#->8t+3a=11#