Viral question from China

This was shared as a primary school question in China, which I gather is at most year 6 and students aged 11-12 years old. Can you solve it? Thanks Blake for the suggestion!

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[LLT]

I’ve taught primary school in China. This is not the kind of question they have to deal with. Circles and geometry, yes; but not this.

[twang]

This is a classic trick question on social media that's faked as a "primary school question" because it looks simple at first. But no even grade 6 math competitions in China will not have arc tangent. Plus that using a calculator is not allowed for primary school exams anyway. However, apart from the calculating arctan(2) part, the rest is indeed primary school level, but as a bonus question that's only meant to be solved by top students.

[SuparnoBhattacharyya]

I wasted 3 hours trying to solve this problem from the perspective of a primary school student (by not using calculus), believing that a simple solution exists. 😢

[flesby]

Since I have relatives from China, I asked them: They think it is highly unlikely that this is a questions asked in primary school. They confirmed that primary school is up to the 5th or 6th grade ... depending on where you go to primary school in china, but even for 6th-graders they said this question is to hard. Calculating with squares and circles to a certain degree yes, but not to this level of complexity. - They think it is more likely that the phrase "Primary school in China" was written beside it, to cater to prejudice that all Chinese people are top notch when it comes to math, which they are not.

[KMR-]

I aced this very problem when I was in kindergarten, the answer is "the area of the overlap of the circles is the blue one".

[JohnnieWalkerGreen]

1) I recall solving a similar problem in the High School exam. However, we were just requested to discover areas that can be calculated (in circle or triangular parts). The expected answer was like:
MIN < AREA < MAX

2) That was the era when logarithm, sine, cosine, etc., were looked up from a table or a sliding rule.

[nilsonrobertoabadias]

I teach physics in Brazil's public schooling and students often have a hard time with simple trigonometry problems by the age of 15-16. I have a hard time believing a problem like this would even be proposed as a university entrance exam

[verkuilb]

I’m surprised that Presh didn’t solve it using multiple techniques (like he often does), and then show how to do it using calculus…

[micomicomi]

Guys I'm from China and I've never seen such hell in primary school. The hardest stuff i got was probably solving system of equations in 5/6th grade. So i assume this question is designed for teenage math olympiad?

[SNOWgivemetheid]

we dont all have the same primary school 😂

[rajatgupta]

Arc tangent being used in Primary school...this question is tough even for high school students 😢

[justalejandro]

I would consider taking two integrals and calculating the difference between them, rather than trying to find the solution with basic geometry

[verkuilb]

How to solve this problem WITHOUT trig or calculus:

Cut a square piece of plywood. Weigh it. Use a jigsaw to cut the two arcs. Weigh the resulting shape. Finishing weight divided by starting weight is equal to the answer divided by 16.

[mrvzhao]

This may not be the best example of a tricky primary school question due to the involvement of trig functions, as others pointed out. However, I did see a friend asking for help on social media about a similar problem that was supposed to be their 4th-grader's homework. It features cleverly constructed geometric shapes with multiple overlapping areas and, like this question, essentially requires the student to uncover hidden quantitative relationships among the various parts. A smart student would be able to visually or verbally articulate the relationships and solve the problem using basic algebra (she may not have learned to formally use variables and will be doing it implicitly using words/drawings). Calculating the result was trivial, so long as the student knows area formulas for basic geometric shapes. That was a much better question, in my opinion, if you want to figure out who the smart cookies are.

[Keldor]

If this is indeed a primary school question, then I'd think the students were learning about approximations rather than trying to give an exact result. Figure out that the square has an area of 16, note that the blue area is somewhere around 1/4 of the area of the full square, and give an approximation of 4 or so for the final answer.

[eyeofregret]

I used calculus to solve and it seemed quite smooth

So the equations of the circles are x^2 + (y-4)^2 = 16 and (x-2)^2 + y^2 = 4
Then find the intersections by isolating y and making x the variables, and you get x = 0 and x = 3.2
As you transformed the circles to functions, the equations are y = sqrt(16 - x^2) - 4 and y = sqrt(4 - (x - 2)^2)
Then integrate from 3.2 to 0 using the following equation,
A = sqrt(4 - (x - 2)^2) + sqrt(16 - x^2) - 4
And you get

There was no way I could've done it the trigonometric way, that was just. Insane

[jesan]

I cheated a bit here to find the relevant integral and ended up asking wolfram alpha for "integral of sqrt(4 x - x^2) - (4 - sqrt(16 - x^2)) from 0 to 3.2", and this produced the correct answer.

[tonyennis]

It's a straight-forward calculus problem, though it comes in four parts:
1. find the intersection of the circles at about (3, 1.5)
2. find the area of the large circle under the curve from 0 to 3-ish (edit - and subtract it from the enclosing rectangle (4*3-ish) to find the crescent area at the bottom.)
3. find the area of the small circle under the curve from 0 to 3-ish
4. subtract those values to find the answer

I can believe a handful of 6th graders could do this - China has a HUGE population and there are exceptional people in abundance for this reason. I do not believe their bog-standard 6th grader would even know the words.

[zainabkhan-yfvt]

Take 2 equation considering a coordinate plane

A circle with centre (0, 4) radius is 4
And a circle with centre (2, 0) radius 2

Use the standard eqn of circle
(X-h)² +(Y-k)²=r²

Find point of intersection

1st point is (0, 0)
As seen in figure

Now we use a concept of calculating area under curve using calculus(integration)

[thredripper]

This question can also be solved by two easier ways,

1- using area of lens formula
2- using coordinate geometry, area under the curve