A nice mental math challenge

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[chinareds]

"Solve using only mental math." Proceeds to show a solution that requires writing out a page of equations.

[universalphilosophy]

Mental solution.

1. Subtract second equation from the first and we see that the difference between the numbers is 4.
2. Now we need two numbers that are 4 digits apart and whose product is less than 54.
3. Square root of 54 is about 7.
4. Hence we start with 7. But we find that as per first equation, 7 + 7 × (7-4) is just 28.
5. Hence we try the next number 8. But 8 + 8×(8-4) = 40.
6. When we try with 9, we get 9 + 9×(9-4) = 54.
7. Hence the numbers are 9 and 5 and the answer would be 9+5+9=23.

[ledebuhr]

Presh,
For your next video, could you interview the 3rd grader that solved this in their head?

[Barreloffish]

I thought mental math means solving the problem in my head without writing down anything. Well, I had a hard time with that approach...

[carton]

Because it said mental math I just sat there for a minute and came up with 23, It’s a shame though since I didn’t realize the negative answer of -22 as I didn’t write down a system

[ok-qrke]

Subtitles:
"hey this is precious tell Walker"

[aijazahmad]

Answer should be 23 as the question is referring to quantity of hats and pies which can't be negetive

[johnnastrom]

Ummm…. it said do it mentally so I assumed that meant I do not write anything down. If you look at the first equation, you can logically conclude that that hat is a factor of 54. 9 is the only number that works and pie = 5 follows. The solution presented here is an example of a complex way to give a solution when complexity is not needed. Also, I doubt that the average 7-yeaar old can solve this.

[aliasmask]

At first, I was going the quadratic route, but it said mental math, so I simplified it. From the first 2 equations you can easily see that hat is 4 more than pie and that pie * hat needs to equal a value near 50 while also being 4 away from each other (like 4 * 8 = 32, too low. 6 * 10 = 60, too high, so must be 5 * 9 = 45). Luckily 45 + the low term equals 50 making 5 = pie and 9 = hat. Plug in 2 hat + pie and get 23. I didn't go negative because you can't have negative pies or hats.

[viksra]

3:15 "we can write [...]" so... in other words, can't be done in your head? So much for mental math!!

[jimdecamp]

Surprised there aren't any comments yet. I didn't look at the video. This is a problem where "mental arithmetic", naive trial and error, is probably easier than the quadratic equation. It's easy to see that hat is four more than pie by subtracting the second equation from the first. The right hand side of the first equation equals 54 which suggests factors like 9 and 6. Try hat = 9, pie = 5 first, and bingo.

So the naive solution is {9, 5}. If you go through and apply the quadratic equation the other roots are hat = -6, pie = -10. I am not sure how one would find this solution by "naive" methods.

[agusmariono]

Negative is not the answer because they dont teach integers in gr 3

[iamyoda]

Wait, Presh. Your solution is *not* fit for a 7 year-old. 99.999% of 7 year-old kids cannot understand your explanation.

[williamallen]

Perhaps more quickly and how I did this in my head was notice that the pie x hat = 54 - hat and also equals 50 - pie.

From this, the two things must be equal so the hat must be 4 more than the pie.

Looking at the top equation, the fact there are two hats with one being a product, and this being equal to 54 suggests that they will be low valued integer solutions, on the thought of 54, 9x6 might spring to mind, and taking off another 9 leaving 45 leaves you with 5 left for the pie to make up the 50 in the second equation. Hence by inspection the pie and hat must be 5 and 9 respectively, meaning the final sum is 23.

[marcosreal]

Grade 3 logic:
1. Can we fix the first two equations so that the order of their items matches?
2. How much bigger is the first equation? (four)
3. How else are the two equations different? The only difference is that the first equation adds a hat instead of a pie. So the hat must be four bigger than the pie. In other words, the hat and the pie are four apart. Let's make a list of some numbers that are four apart: 1 and 5, 2 and 6, ...
4. Let's look at the first equation again. Notice that part of it is the hat times the pie. Remember that the hat and the pie are four apart. Are there any numbers in our list of numbers that are four apart that multiply to a little bit less than 54? 10x6 = 60 is too large. But 9x5=45 will do. So lets's put hat=9 and pie=5 into the first equation and see what happens. It works!
5. From here we find the required value of 23.

(In grade 3 they haven't yet studied negative solutions But you could change the first equation to subtract the hat and the second one to subtract the hat and then solve this new problem using similar logic).

[abhinavdiddigam]

If it was given to me when i was 7 yr old, I would have looked at the pie and said when do I get to eat this

[rasowa]

The answer is: YES, I can solve it using only mental math.

[the_luna_lily]

Paused the video
Got a piece of paper
Worked it out
Played the video
‘Using mental math’
Fish frick!

[girivkodur]

If we were to guess positive integers, we could also rethink the equations as
h(p+1)=54
p(h+1)=50
Then think of integer solutions which satisfy the equations. I think that would be much faster. IMO

[Epiciouss]

I just found the root of the quadratic equation: p^2 + 5p - 50 and found the answer. Technically same as your explanation since the basis on how to figure the root is to find the factors of 50 which sums to 5.