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?

Question

Eli Assouline · Accepted Answer

8 tetra and 3 angel fish

#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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #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#

Jameson Allen · Answer

undefined Let's denote the number of angelfish as $ A $ and the number of neon tetras as $ N $.

The constraints are:
1. Each angelfish costs $12 and needs at least 6 gallons of water.
2. Each neon tetra costs $3 and needs at least 3 gallons of water.
3. The total cost cannot exceed $60.
4. The total water capacity cannot exceed 30 gallons.

The equations representing the constraints are:
1. $ 12A + 3N \leq 60 $ (total cost constraint)
2. $ 6A + 3N \leq 30 $ (total water capacity constraint)

We also have non-negative constraints: $ A \geq 0 $ and $ N \geq 0 $.

Now, we can solve these equations to find the possible values of $ A $ and $ N $.