I encountered this problem:
Farmer Joe needs to buy tools. He can go to two different stores, but only wants to go to one. At Lowes, 5 shovels and 12 clippers are 14.58. 6 shovels and 14 clippers are also 16.50 at Lowes. At Home Depot, a shovel is $1.89 and a clipper $0.42. Joe needs to buy 8 shovels and 20 clippers, and wants the cheapest option. Where should he buy his tools from?
In this problem, is the information about how much 6 shovels and 14 clippers is in total necessary to solving this solution? Because in my mind, I found the price of 5 shovels and 12 clippers from Home Depot and compared it with the price of 5 shovels and 12 clippers from Lowes to get my answer. Is my way of thinking erroneous?
Consider a related but different problem where there are two stores. The store "One Dollar Store" where everything costs one dollar, and the store "Everything Free But Shovels" where everything is free except shovels and those cost two dollars each. From here on I'll just call these stores store $O$ and store $E$ for short.
We look at the total cost of buying two shovels and four clippers from each store and we find that it cost a total of six dollars from store $O$ and it costs a total of four dollars from store $E$. For this particular purchase then, store $E$ has the lower total price.
Now, what if we were to purchase three shovels and two clippers? Well, store $O$ would charge us a total of five dollars while store $E$ would charge us six dollars. For this purchase, store $O$ has the lower total price.
Despite that store $O$ had a more expensive final bill for the first particular purchase than store $E$, having changed the amounts of things being bought store $E$ happened to have the more expensive final bill the second time around.
The same thing could potentially happen with your original problem. Even if you were to compare which store had the cheaper total cost of the specific purchase five shovels and 12 clippers, this does not give you enough information to know which store would be the cheaper option for buying eight shovels and twenty clippers.
Given the information in the problem, you should be able to determine exactly how much an individual shovel costs at Lowes as well as exactly how much an individual pair of clippers costs at Lowes. You are already told this information for Home Depot. You can then using that information determine the final total for the bill of our desired purchase from each store.