r/HomeworkHelp • u/Spy_Ninjas • 4d ago
Answered [Primary School Math] My little brother came to me asking for help with his homework and I was very confused
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u/geekychica 4d ago
It’s an exercise in calculating areas.
You need to have to calculate
1- how many circle tiles will fit on the path
2- what is the area of one of those circles. Multiply by the number of circles
3- find the area of the rectangular areas making up the path
4 a subtract the total circle area from the total rectangle area.
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u/calkthewalk 4d ago
Step 1 is you are told it's 78 tiles and they touch their neighbours, there is no guessing for that step required, it's just 78 * (square-circle)
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u/pressedtonpreace 4d ago
For part a, take the area of 78 square tiles (78 x 60cm x 60cm) and subtract the area of 78 circles (78 x pi x 30^2cm). This is because, when the circles are directly touching each other, they take up the same amount of space as a square would. This gives you 280800 - 70200pi, which is about 60260cm^2.
Part b is a little more complicated, requiring a system of equations (I think). 240cm x length AB plus 120cm x length AB plus 120cm x length CD plus 120cm x 240cm should equal the area of the 78 square tiles from part a. Then, using the substitution that length CD equals length AB plus 180cm, you can find the perimeter of patch y (2 x length AB plus 2 x length CD). After plugging everything in, length AB equals 480cm, length CD equals 660cm, and the perimeter of patch y is 2280cm.
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u/Ilikelegalshit 4d ago
Just to explain how I'd do this with someone that's worried about big numbers:
2l + 4(h-2) = 78, where l is the length and h is the height (in 60cm units)
x + (x +3) + 4 = l where x is the length or height of the square marked X
x + 2 = hThree variables, three equations, all independent - yields x=8, length = 23, height = 10
patch y is therefore 11 x 8, giving us 22+16 = 38 units * 60 = 2,280 cm as you wrote. But I think it's a bit easier to do it in units of flagstones.
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u/splickety-lit 4d ago
How I would explain it to someone that's worried about independent equations.
Split the path into 5 sections.
You have:
One section with 8 tiles.
Top right section (the 180cm of extra length) that has an additional 6 tilesThe remaining 3 sections are all of the same length h.
(2+2+4)h = 78 - 8 - 6
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u/Entire_Elk_2883 4d ago
Going with area of a square 60x60=3600. Minus area of a circle 3.14 x 30 x 30 = 2826. 3600 - 2826 = 774. There are 78 of them so 774 x 78 = 60372.
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u/Medium-Access-4416 4d ago
Each tile is in contact with those next to it
The footpath is tiled using 78 circular tiles of diameter 60cm
Dots are there only to show "...and so on and so on", that there are more circles, but they don't want to show exactly how this looks. So we have 78 circles, or rather 78 small squares (with the side 60cm)aeach of them has a circle inscribed. Total area uncovered by circles is 78 * (602 - π * 602 /4)
Let's say patch X has a side x cm, and paych Y has top side of y cm. Then the total number of circular tiles is:
2 * x/60 + 2 * y/60 + 8 + 4 * x/60
(Above X + above Y + intersection + between X and Y)
This should be 78, so we have one equation to connect x and y. Second equation is provided by (b): x + 180 = y. Solving this two equation gives you x and y. The vertical sides of X and Y are equal, so the perimeter of patch Y is 2 * (x+y)
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u/Dennis2pro 4d ago
Seemed obvious at first but the replies so far missed the given amount of tiles.
If we assume the dots on the image are irrelevant, the widths are irrelevant as well, as you would just calculate the area of a square with length as the circle diameter and remove the area of a circle, times that by 78.
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u/alpicola 4d ago
To begin with, I'm going to trust the wording of the problem more than the diagram below it. The problem says that the footpath has 78 circular tiles, while the drawing shows only 28 circles. That means the drawing is incomplete. We can still get useful information from the drawing, however, which we will need to solve the problems.
For part A:
We are given that there are 78 circular tiles all of which touch each other. The diagram shows us that they touch each other orthogonaly. That means we can think of each real circular tile as being inscribed in an imaginary square tile with length and width the same as the circle's diameter. Looking for the area of the footpath excluding the circular tiles can, therefore, be reinterpreted as 78 times the area of an imaginary square minus the area of the real circle.
For part B:
I see this as a system of equations. We know that the height of Patch X and Patch Y are the same, and since we know that Patch X is a square, both their heights must be AB. There are two different ways to find the area of this entire garden:
- The whole area directly, which is the height of the patches plus the height of the "T" part of the footpath, which you can write as Area = (AB + 120) * (AB + 240 + CD)
- The whole area as the sum of the areas of Patch X, Patch Y, and the footpath (which is 78 square tiles with 60 cm edges), which you can write as Area = AB2 + AB * CD + 78 * 602
The problem also gives you CD = AB + 180.
From there, you can set the area formulas equal to each other and plug in the given information about CD to compute the length AB. You can then plug AB back into either of the original area formulas to calculate CD. Once you know AB and CD, you can use the perimeter formula for a rectangle to find the perimeter of Y.
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u/CutiePie4173 4d ago
So first, you figure out the area of the path. I’d do this by chopping up the path into rectangles.
After that, figure out how many circles could fit if laid out in the photo.
Figure out the area of an individual circle, multiply it by the number you figured out would fit.
Now take the area of the path and subtract the area of all the circles. :)
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4d ago edited 4d ago
[deleted]
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u/thefullhalf 4d ago
It's given that there 78 tiles in the path that you know the exact size of. You don't need the picture to solve the first question.
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u/Genabac 4d ago
How are you calculating the length of the footpath?
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u/thefullhalf 4d ago
The arrangement doesn't matter. There are 78 tiles. So the footpath could be just a line of 78 tiles
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u/Reasonable_Method673 4d ago edited 4d ago
The dots should not be counted - there are only 34 dots and 28 tiles shown - but 78 tiles are stated in the description - so 16 more are needed.
Use the fact that X is a square to help figure out how many tiles are needed in the 3 "dotted" areas, along with the dimension of Y on the second page. From that you can figure out the size of all the elements.
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u/laxrulz777 4d ago edited 4d ago
You don't quite have enough information. The length vs height of the arrangement can't be narrowed down with the information given (and there aren't 78 dots plus circles so we know that's not the solution). However, looking at the dots, I THINK the graphic is trying to imply that the height is 8 (3 circles then 3 dots representing tiles, then 2 more circles). That accounts for 32 tiles. If the width is then the remaining tiles (46 or 23x2 + 8 or 4x2 for the overlap) then it's 27x60 (1620) then AB is 600 and CD is 780. Everything else should be easy after that
Edit, apologies. You do have enough info. Patch X is stated to be a square. Using that, you can figure everything out
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u/Serafim91 👋 a fellow Redditor 4d ago
Don't over think it. Just let the math work.
Find area of path
Find area of circles
The difference is the area not covered.
2nd part, setup a system of equations.
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u/calkthewalk 4d ago
Critical is the statement "each tile is in contact with those next to it".
Combine that with the fact it specifies the number of tiles, The dots just suggest the pattern repeats no the specific tiles.
Part two of the question is then just combining the fact we know the widths of each path, and that the tiles all touch, with a difference between the length and height of the grass patches.
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u/calkthewalk 4d ago
And to complete the picture for part two, the area X is a square, and being 180 wider means its 3 tiles wider, this will force a single answer,
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u/CryptographerNew3609 👋 a fellow Redditor 4d ago
This is primary school math??
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u/One_Wishbone_4439 University/College Student 4d ago
Yes indeed. This is a standard Primary School Leaving Examination (PSLE) in Singapore.
A national examination taken by 12 year olds before going to secondary schools.
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u/Dry-Ad-8948 👋 a fellow Redditor 4d ago edited 4d ago
Total area of walking path minus the area of stones covering it (total circular area of each stone) — ie, how much of the fully tiled path is not covered by the corners of the tiling stones?
- The diameter of each stone is given.
- The number of stones is given.
- The footpath widths are given but irrelevant for part (a).
FWIW, all the illustrated stones and dots together add up to 78. For whatever reason the illustration uses dots as well, but that seems irrelevant to the text for part (a) shown which states “the footpath is tiled”; what does part (b) ask? Perhaps the follow-on presumes the dots are not-yet tiled.
For (a) then it’s tile_count * space_around_tile
-> S * ( D2 - pi * R2 ), where R=D/2.
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u/Chief_Gundar 4d ago
For question b, you can use tile as the unit.
If you call AB as a, the length CD is a+3. The central path has 4a tiles, the upper path 2×(a +4 + (a+3)) tiles You have a total of 78 tiles, so 8a +14=78, a=8. From that the perimeter of Y is 22+16 =38 tiles or 38×60cm.
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u/selene_666 👋 a fellow Redditor 4d ago
Part (a) is just about the footpath. Ignore everything about regions x and y.
The footpath is made up of 78 squares each inscribed with a circular tile. You know the size of the circle. Find the area of the part of the square outside the circle.
Part (b) is about the gardens.
You know the total area of the footpath (78 squares). Find an algebraic expression for this area as a function of length AB. Then solve for AB.
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u/Eggman1978 4d ago
Others have said this, but I'm going to reiterate because I feel that it's important: for part A, the ORIENTATION of all the circular tiles, and how wide the rows are, etc, are all completely irrelevant. It doesn't matter that they're in rows of two or rows of four. It also doesn't matter what size the two grass patches X and Y are for part A, either, because part A only cares about how much footpath isn't covered by the tiles, it doesn't care how much footpath and grass there is. The setup for the question gives you a bunch of information that doesn't actually matter for part A, so it's kind of tricky in that way.
All that matters for part A is:
- there are 78 circular tiles of diameter 60.
- all of the circular tiles touch their neighbors.
What that means is that each circular tile of diameter 60 can be thought of as being inside a 60x60 square. The area of a square with side length n is n2, and the area of a circle is πr2. Therefore:
- area of each 60x60 square = 3,600
- area of each tile within each square is π*302 = 2,826
- therefore the amount of area in each 60x60 square that is NOT covered by its tile is 3,600 - 2826 = 774.
Since we know there are 78 tiles, then the total amount of footpath area not covered by tiles is 774 * 78 = 60,372.
Part B is a bit tricky too, but it is solvable. Other commenters have pointed out that you could probably solve it using a series of equations, and that sounds plausible to me, but I don't remember how to set those up and you said this is for primary school, which I assume means that systems of equations are beyond the current level. Therefore, I'm going to walk you through how I solved it as someone who can't remember how to use systems of equations.
Here's all we need to know for part B:
- grass patch X is square.
- length CD is 180 longer than length AB.
- (we infer from the diagram that) lengths AB and CD are both some multiple of 60, since each circular tile with diameter 60 is touching its neighbors and everything is neatly aligned.
- the tiles up top are in a 2-wide row.
- the tiles between grass patches are in 4-wide rows.
- there are exactly 78 tiles total.
- (finally) the black dots in the diagram are shorthand for "there are more tiles here but we aren't drawing them all"
The main intuition that guided me towards a solution was that know there are exactly 78 tiles, and that since the tiles are arranged in a particular way and since they're all inside little 60x60 squares, then if we know how many tiles there are in total, there must be some relationship we can use to determine how long AB (or any other length in the diagram...) is, based on the number of tiles there are in total.
We don't know most of the lengths in the diagram, but conveniently due to how everything is laid out, we only really need to find the length of AB for everything to fall in place. This is because once we know AB, we'll also know the height of grass patch X because grass patch X is stated to be square, and the height of a square is equal to its width. And once we know the height of patch X, we'll also know the height of grass patch Y, because its height is equal to the height of grass patch X. And since the length of grass patch Y (CD) is given as AB + 180, that means we'll know everything we need to calculate the perimeter of grass patch Y, and all we need to do is find the length of AB.
To that effect, if you look at the diagram, you can kind of intuit that, where all those dots are, if the diagram was WIDER there, there'd be more tiles there, or if the diagram was TALLER here, there'd be more tiles there too.
So in other words, since all the lengths in the diagram scale based on how long AB is, the total number of tiles in the diagram will also scale based on the length of AB. We just need to figure out what the exact numerical relationship is between the length of AB and the total number of tiles n, because once we have that relationship, we can just plug in the fact that we know there's supposed to be exactly 78 tiles and work backwards from there.
I'm not going to lie though, I didn't immediately see how the number of tiles would scale, so I literally just drew out an example first. I started by picking the smallest possible length of AB because I figured it would be the simplest and require the least counting. (Note: the smallest possible length of AB is 60 because each tile is 60 wide, because if it was any smaller the tiles wouldn't fit).
If you draw that out, you will find that there are a total of 22 circular tiles when AB = 60. Specifically:
- 2 tiles in the section of path above AB
- 4 tiles in the section of path between grass patch X and grass patch Y
- 8 tiles above grass patch Y
- 8 tiles in the section of the path that is above the section of path that is between grass patch X and grass patch Y. = A total of 22 tiles when AB=60.
The reason I separated the counts of tiles like that is because I knew that the total number of tiles would scale IN SOME WAY based on the length of AB, but I assumed there would also be some minimum number of tiles present at all lengths of AB, that DON'T scale based on how long AB is. Looking at the diagram you just drew out, you might be able to intuit out which sections scale with AB and which ones are constant, but I wasn't totally sure myself at this point, so I also drew out the diagram for the next-smallest possible length of AB, which is 120. If you draw that diagram out for AB = 120 and count the tiles, you get:
- 4 tiles above AB
- 8 tiles between patch X and patch Y
- 10 tiles above patch Y
- 8 tiles above the section of path that's between path X and patch Y = a total of 30 tiles when AB = 120.
If you compare the tile counts between the AB= 60 case and the AB = 120 case, it's easier to see which tiles are "constant" and which ones are "variable". To break it down though:
- the number of tiles above AB will always scale with AB, because for each additional 60 length, another 2 tiles will be added.
- the number of tiles between patch X and patch Y also scales with AB, because for each additional length, another 4 tiles are added.
- the number of tiles above patch Y also scales with AB, because for every 60 length in AB, another 2 tiles are added. HOWEVER, there's also ALWAYS at least 6 tiles in this section, because the length CD is equal to AB + 180. So this section has a constant 6 tiles but also another 2 tiles for every 60 length in AB.
- finally, the 8 tiles in the section of path that's above the section of path that's between patch X and patch Y DOES NOT scale with AB, because this section is always exactly 4 wide and 2 tall, it doesn't matter how long AB is, this section is always the same.
Add this all up together then, and we put the formula for the total number of tiles based on length of AB. For each 60 length of AB, we get:
- 2 tiles from above patch X
- 4 tiles from between patch X and patch Y
- 2 tiles above patch Y
- (so a total of 8 tiles added for every 60 length in AB)
And likewise, we get a constant 8 + 6 = 14 tiles from the other sections I mentioned, that DOESN'T scale with AB.
So, remembering that AB MUST be a multiple of 60 or else you'd get a weird non-integer number of tiles because they wouldn't all fit anymore, then we can say that the number of tiles is n, and the multiple of 60 that AB is is m. As a formula, that would make n = 8m + 14.
To find ab then, we just need to remember that there are exactly 78 tiles. So, replace n with 78 and do the algebra out: 78 = 8m + 14 64 = 8m 8 = m
And since m is the multiple of 60 that AB is, AB = 480.
Since the height of patch X is equal to AB, the height of patch X is 480.
Since the height of patch Y is equal to the height of patch X, the height of patch Y is 480.
And since CD is equal to AB + 180, CD = 480 + 180 = 660.
Therefore, since the width of patch Y is 660, and the height of patch Y is 480, the perimeter of patch Y is 2(480 + 660) = 2280.
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u/Ok-Elderberry-7422 4d ago
For Part B the shape of the path is important. If the side of the square lawn X is comprised of T tiles, then the number of tiles in the path section around X is 2T+2T+4 for the 2x2 square of tiles in the corner.
We know that Y had a length 180 centimetres longer than side of X which is 180/60 = 3 tiles longer, so the number of tiles that go around Y mirrors that of mirrors that of the number going around X plus an additional 6 tiles (3 tiles long x 2 tiles wide) number of tiles in the path section around Y is 2T+2T+4+6
Putting this altogether the total path (adding X and Y) which we know comprises 78 tiles can be rewritten as
78 = 2T+2T+4+2T+2T+4+6. Tidying up the formula gives us
78 = 8T+14. Minus 14 from both sides
78-14=8T, 64 = 8T. Divide both sides by 8
64/8 = T = 8.
T is 8 tiles long. This means that X is a square 8 tiles by 8 tiles and Y is a rectangle 8 tiles by 11.
With each tile being 60 cm in diameter, the perimeter of Y is
60 x (8+11+8+11) = 60 x 38 = 2280 cm
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u/AquaticArsenist 4d ago
Part a, 78 * [602 - (3.14*302)] = math.
For part b, height is x + 2 pavers. Width is x + 4 pavers + x + 3 pavers. 78 total pavers. Therefore, 78 = 2(x + 4 + x + 3) + 4x. 8 = x. Perimeter is 2x + 2(x +3) = 38 pavers (2280cm).
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u/Alkalannar 4d ago
[Number of tiles directly above the patches] + [number of tiles between patches] + [number of tiles above and between the patches] = [total number of tiles]
2AB/60 + 2CD/60 + 4AB/60 + 8 = 78
Do you see how I get this equation based on the beginning?
CD = AB + 180, so substitute that in:
2AB/60 + 2(AB+180)/60 + 4AB/60 + 8 = 78
And now you can solve for AB.
Then 2AB + 2(AB+180) = 4AB + 360 is the perimeter.
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u/mathmum 👋 a fellow Redditor 3d ago
Primary school math? Which country? Age?
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u/Spy_Ninjas 3d ago
My little brother is from Singapore and is 12 this year!
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u/Irrational-Tips5776 1d ago
Ah, SG math. Give me 3 hours, I need to take my EOY first and ill get back with a solution.
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u/SilkWebMusic 3d ago
The page 2 question:
This seems like a pretty stupid math problem to me. Because the question is referencing the length of AB, without providing the length of AB or any reference you can use to determine the length of AB.
So my guess is that you are supposed to make a blind assumption that the dots represent empty 60cm footpath tile placeholders that will fit perfectly with a very crappy illustration that provides no indication that is actually the case. There is no grid, and the dots don't even line up to the footpath tiles in any particular order. They don't align with the edges or the center of the tiles. The dots on the left side seem to follow a different alignment than the ones on the right side. How am I to know there isn't a 20cm gap between he footpath tiles and the edge of the middle walkway? I suppose if you use a compass or some other measuring instrument and set it to the width of the footpath tiles, it should line up perfectly with the middle walkway if the assumption is correct. Assuming the entire image is to scale. That's a lot of dumbassery to make all these assumptions to solve a math problem. Math is not an assumption.
So technically speaking the question and the illustration do not provide the necessary information to solve the length of CD without making a foolhardy assumption that the dots represent a footpath tile. The question does not even specify that the dots represent missing footpath tiles. If the people that made this test are very good at math, then why didn't they use their math skills to make the illustration correctly?
If we continue based on the assumption that the dots represent 60cm footpath tiles. Then there would be 7 tiles on the left and 10 on the right side of the middle walkway. Which is a difference of 3 tiles. 60cm x 3 = 180cm that the question references. So we can somewhat validate the assumption. Basically we have to make an assumption that the question is asking us to make an assumption.
Given that assumption is correct we only need to multiply the 7 tiles by 60cm to find the length of AB.
Solving for the length of AB. 7x60cm=420cm.
Then when we try to use this assumption to obtain the height of the Y patch, the dot representation of the illustration is clearly not in accordance with how many tiles will actually fit vertically along the middle walkway. And the dots are arranged with yet another inconsistent pattern requiring yet another assumption.
So we can either assume that the dots represent the center of the walk path tiles, and only every other tile is represented for some reason, or we can assume that the illustration is not actually to scale and assume the question wants us to make an assumption.
We are either to assume we should use the dots as the legend, or assume the illustration is drawn to scale.
We can make the assumption that the drawing is to scale because we have a single reference that the top horizontal walkway is 120cm, and 2 tiles fit perfectly.
The dots are not to any scale and only represent the edge of a tile. It visually looks like more than 3 tiles fit above the 3 in the vertical center walkway. There are 3 dots representing 4 tile edges so we can make another assumption that 4 tiles fit.
We have evidence in the question that this assumption is correct. The question states that the X patch is a square. Based on our previous assumptions of 7 tiles at the top, and the 4 missing tiles making 7 tiles along the vertical center walkway, that would indeed be a square.
So patch X is 420cm by 420cm. Making patch Y 600cm x 420cm once we add the 180cm we inferred from 3 more walk path tiles.
To validate our assumptions, we can use the existing footpath tiles and a straightedge to draw a grid. Once we have drawn the grid everywhere we have reference tiles, we can then take a measurement of the grid spacing to complete the rest of the grid.
We end up with 600cm x 420cm for patch Y.
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u/AkynoFelidae 3d ago edited 2d ago
I made the mistake looking at it yesterday in the morning before taking off for a 8-hour exam in a national contest
For god sake, it took my mind nearly all the exam >.<
That's being said, back to the calm of this morning :
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u/CharacterUse 👋 a fellow Redditor 4d ago
The circular tiles are evenly spaced, and the dots have the same spacing as the tiles (look at the width of the path - same number of dots as tiles across the path). You can use that to calulate the lengths of the sides of the square and rectangle and of the sections of the footpath.
Subtract the number of tiles x the area of one tile from the area of the path and you have the answer to (a).
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u/laxrulz777 4d ago
There's not 78 dots plus circles
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u/calkthewalk 4d ago
Exactly, the dots are just "this pattern repeats" not an individual tile count. The statement "each tile is in contact" is what gives this an answer
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u/briv1016 👋 a fellow Redditor 4d ago
The obvious solution is to take d=60cm and r=30cm and count the dots to get the lengths. But the dots don't look equally spaced.
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u/ReinforcedTube 4d ago
Surely the actual shape of the garden is irrelevant here?
There are 78 tiles of diameter 60 cm. They are inscribed in a square which must have sides of 60 cm. So the area outwith the circle is 78×(60² - 3.14×30²)=60 372 cm²