This is my son’s 6th grade math problem. We figured out the answer to be 324 peanuts but only by drawing a picture and working backwards. What’s the best way to solve this? Is there an equation for a problem like this?

I don't know if you're doing algebra yet...

Remember that when you remove a fraction, you're left with (1-fraction), and that a fraction of a fraction is multiplication.

So if you started with x peanuts, you end up with (1-1/3)(1-1/4)(1-1/2)x-10  ...

(2/3)((3/4)(1/2)x-10 = 71

6x/24 -10= 71

X/4= 81

X=324

But working backwards is a perfectly acceptable method.
Working backwards is IMO a great way to do it, and probably the most direct path.  It is how I would do it for sure.  I'd even argue it's hardly backwards when you are going from what they gave you (remaining peanuts) to the answer (starting peanuts).  And it can totally be done formulaically once you work out what operation 'undoes' taking some fraction of the peanuts.  That's pretty clear for taking 1/2 (multiply by 2), but what about taking 1/4?  Well that means the peanuts left was 3/4 of what was there before.  The amount of peanuts taken (1/4 of the original) is then 1/3 of the peanuts left (3/4 of the original).  So you add on 1/3 of what you had, or in other words multiply by 4/3.  You can also think about that by saying you got to what you have left by multiplying by the 3/4 remaining after taking 1/4, so to get what you started do the opposite and divide by 3/4 (equivalent to multiplying 4/3).  Once you see how to do that it is pretty quick to write out the formula:

(71+10)\*2\*4/3\*3/2 = 324

That said the way the other commenter did it by first writing out the situation as described with a variable:

x\*(1-1/3)\*(1-1/4)\*(1-1/2)-10=71

and then using the consistent and established rules of algebra to get the answer from there is also good (if you do that you might see you end up doing exactly what I described in the first section anyway, add 10, and then divide by 1/2, 3/4, and 2/3). But doing it algebraically like this may leave potentially less room for error (and if there is an error, a more detectable one) as it clearly delineates the process of modelling the problem in the language of math and then solving the problem using a clearly defined mathematical toolset rather than sort of doing both as you go working backwards one operation at a time.

Ultimately there are many ways to go about it, and IMO one of the beauties of math is that as long as you know and understand what path you are taking, you'll arrive at the same answer as someone who might have seen and worked the problem with a completely different line of reasoning.  The path you take to the answer is really much less important than building a good understanding and framing of the problem where you are confident in your reasoning toward that solution.
Least common denominator is 12
Phil gets 4/12
Joy gets 2/12
brent gets 3/12
3/12 remaining = 81.
1/12= 27
Total peanuts is 12x27 = 324
Phil gets 4x27= 108
Joy gets 2x27 = 54
Brent gets 3x27 = 81
Honestly working backwards and drawing a picture is a pretty good way to do it. The best way to solve a problem is the way that logically makes the most sense to you and is mathematically sound
There's no real need for algebra, and working backwards is perfectly fine here.

This is (71 + 10) ÷ (1/2) ÷ (3/4) ÷ (2/3) = 81 x 2 x (4/3) x (3/2) = 81 x 4 = 324.

You can cancel the 2's and the 3's in multiplying together 2 x (4/3) x (3/2).
Simplify first. I'd add the subtracted amounts up to make it one fraction, then add those to 71, then answer questions a and b. Fast work.
(2/3*3/4*1/2)x - 10 = 71

x = 324
Yes, what I did was first add the 10 back, then multiply by 2, then multiply by 4/3, then multiply by 3/2
I find working backwards is the simplest way for me to solve problems like this.

[(Total) * (1-1/3) * (1-1/4) * (1-1/2 )] - 10 = 71

= [(Total) * (2/3) * (3/4) * (1/2 )]  = 81

So,

81 / (1/2) / (3/4) / (2/3) = 324
Algebra solution:

P \* 2/3 \* 3/4 \* 1/2 - 10 = 71 Cancel 2/2 and 3/3

P \* 1/4 - 10 = 71

P \* 1/4 = 81

P = 81 \* 4 = 324
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