Question

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?

Answer 1

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`