I’m cramming for a test and trying to solve simultaneous equations by graph, but with y = 2x + 1 and y = -x + 4 I somehow set the slopes equal (why did I do that?) and then confidently circled (0,0) as the intersection because it looked tidy-what am I supposed to look for on the graph to get the real point?

I keep stumbling over bounds and error intervals when a value is rounded, especially the inclusive vs exclusive endpoint. For example, if a length is written as 8.4 cm rounded to 1 decimal place, what exactly should the interval be? Is it 8.35 ≤ L < 8.45, or do I include 8.45 as well? And does writing 8.40 to 2 dp change anything about the endpoints? I have lost marks before for using ≤ at both ends, so I am trying to understand the rule in a way that sticks. I feel like I am missing one key idea about which values would actually round to the stated number. Related to that, for a simple sum: if two sides are each given as 3.2 m to 1 dp, what is the tightest possible error interval for their total length? Do I just add the bounds for each, or is there a more careful way to do it? I am probably overthinking this, but I would appreciate a clear way to decide the endpoints in these cases.

I’m trying to get my head around bounds and error intervals and I keep ending up in a tangle of brackets and inequalities.

Say I measure a rectangle and write the length as 7.3 cm to 1 decimal place and the width as 4.8 cm to 1 decimal place. I think I know the error interval for each number, but when I try to get the possible area, I freeze. Do I multiply the lower bound by the lower bound and the upper by the upper, or do I need to consider “mix-and-match” combos too? And if the upper bound is supposed to be exclusive, does that make the area’s upper bound exclusive as well?

Related mini-mystery: if something is given as 2.40 to 2 decimal places versus 2.4 to 1 decimal place, do those have different error intervals, or are they secretly the same number in a fancy outfit?

I keep second-guessing which inequalities should be strict and where the extremal values for the area actually come from, so my answers wobble around. Any help appreciated!

I’m trying to clean up my understanding of place value when there are zeros in the middle of a number. I keep feeling confident, then I trip over expanded form or regrouping and second-guess myself. I remember losing points on a quiz because I “simplified” an expanded form in a way my teacher said wasn’t valid, and I’ve been cautious ever since.

Example: with 3,040, I label it as 3 thousands, 0 hundreds, 4 tens, 0 ones. In expanded form I wrote 3,000 + 40. Then I tried regrouping and wrote 30 hundreds + 40, and also 304 tens + 0 ones. Is it okay to say 304 × 10, or is that mixing forms in a way that’s not standard?

Another example: 40,506. I wrote 4 ten-thousands, 0 thousands, 5 hundreds, 0 tens, 6 ones. For expanded form I put 40,000 + 500 + 6, but I also tried two “regrouped” versions: 405 × 100 + 6 and 40 × 1,000 + 506. Are either of those considered correct ways to show regrouped place value, or am I breaking a rule by compressing across a zero place?

More generally, what’s the correct way to handle zeros in expanded form? Do I need to include terms like 0 × 1,000 or 0 × 10, or is leaving them out better? And when I regroup (like trading 1 thousand for 10 hundreds), is there a clean, consistent rule so I don’t accidentally change the value or the form? I’d really appreciate a straightforward way to check myself on numbers like 3,040 and 40,506 because I keep making small mistakes around the zeros.

I’m revising fundamentals and keep tripping over cone volume-if I’m given the slant height and the diameter, how do I get the actual height/radius I need for the formula? Follow-up: if it’s an ice-cream cone filled level to the rim, does that change how I should think about the volume at all?

How can I quickly tell the shifts, reflections, and stretches in something like y = -2*f(3(x-1)) + 4 without mixing up horizontal vs vertical? I’m preparing for a test and tried the ‘inside affects x’ trick and a quick sketch, but I’m not sure if that’s even relevant or if I’m using the right order.

I’m trying to get my head around cumulative frequency, and my graph keeps doing interpretive dance when I just want a nice smooth ogive. I have grouped data with class intervals, and I’m not sure where to plot the points for the cumulative totals. Do I put them at the midpoints of the classes, or at the upper class boundaries of each interval? I’ve seen both in different places, and my estimated median changes depending on what I pick.

Personal saga: on a test last term, I plotted everything at the midpoints and my median ended up way off. My teacher gave me that sympathetic “you stacked your blocks on the wrong shelf” look. Since then, I’ve been nervous about whether my curve should climb like stairs at the ends of classes or hover at the middle like a confused pigeon.

Analogy that may be completely wrong: is cumulative frequency like pouring sand into a series of buckets (so the total should be measured at the rim of each bucket), or like checking the weight at the balancing point (the midpoint)?

Could someone explain, plainly, where the cumulative points should go for a “less than” cumulative frequency graph? When, if ever, are midpoints okay? How do you handle open-ended classes or different class widths so that the median and quartiles you read from the curve actually make sense?

I’m revising to strengthen my fundamentals on negative indices and I think 2^-3 = 1/8 and 5x^-2 = 5/x^2 (love the flip-it pattern!), but I’m doubting whether the 5 should flip as well. If that’s right, how do brackets/fractions play with this-like (2/3)^-2 or (ab^-1)^-1-do we invert the whole thing then apply the power, or am I mixing rules?

I keep tripping over substitution and minus signs. If x = -3, I get confused about expressions like -x^2, (-x)^2, 5 – x^2, and (2 – x)^2 – which parts are actually being squared, and when does the negative sign get squared with it? I thought I understood order of operations, but the little dash in front of x^2 keeps tricking me.

I also get stuck when the thing I’m substituting is an expression. For example, if x = 2k – 5 and the original expression is -3x^2 + 4x – 7, am I supposed to write -3(2k – 5)^2 + 4(2k – 5) – 7 every time, or is that overkill? And what about something like 10 – x when x = 3k – 1 – do I need 10 – (3k – 1), or is 10 – 3k – 1 okay?

Is there a simple way to think about when parentheses are required during substitution so I stop losing minus signs? Any help appreciated!

I’m practicing index laws and I keep mixing up when to add exponents and when to multiply them, especially once negatives and parentheses jump in. The expression I’m trying to simplify is:

(2x^-3 y^2)^-2 ÷ (4x y^-1)^-1

I know the rules in theory: same base multiply -> add exponents, power of a power -> multiply exponents, division -> subtract exponents, negative exponent -> reciprocal. But when I try to apply them all at once, I get tangled.

My attempt:
– For the first part, (2x^-3 y^2)^-2, I wrote: 2^-2 x^(+6) y^-4. I think (x^-3)^-2 gives x^6, and (y^2)^-2 gives y^-4. The 2 becomes 2^-2 (right?).
– For the second part, (4x y^-1)^-1, I wrote: 4^-1 x^-1 y^1. I figured the -1 on y^-1 flips it back to y^1.
– Then I tried to divide: [2^-2 x^6 y^-4] ÷ [4^-1 x^-1 y^1]. For the variables, I did x: 6 – (-1) = 7, and y: -4 – 1 = -5. But for the numbers I froze: is 2^-2 ÷ 4^-1 the same as 2^-2 * 4^1? Or should I be combining them some other way?

Could someone show a clean, step-by-step way to handle this without skipping steps, and point out exactly where my reasoning goes off? I mainly get confused about the coefficients with negative exponents during division, and keeping the signs straight.

Does 1/2 * base * height still work when the triangle’s tilted and the perpendicular height falls inside or even outside? I keep grabbing the “nice” side as the base and pairing it with the wrong length because the real height isn’t a side.

I’m trying to get my head around how Pythagoras works in 3D, and I keep second-guessing myself. For a simple box, say 5 cm × 8 cm × 2 cm, I want the distance from one corner to the opposite corner through the inside. I did it in two steps: first the base diagonal is sqrt(5^2 + 8^2) = sqrt(89). Then I used that with the height: sqrt(89 + 2^2) = sqrt(93). That feels a bit weird though-like I’m taking a square root only to square it again. Is it actually okay to skip straight to sqrt(5^2 + 8^2 + 2^2)? Why does that work, or am I just getting lucky with the numbers?

I also tried the coordinates version. If I have A(1, 0, 2) and B(6, 4, 6), I used the distance formula: sqrt((6−1)^2 + (4−0)^2 + (6−2)^2). I think that’s right, but I’m struggling to see the actual right triangles in 3D that justify it. Like, where exactly are the right angles hiding? Is there a neat way to sketch the two-step triangles (first in a plane, then in 3D) so it doesn’t feel like magic? My mental picture goes a bit squinty here.

Follow-up: what if the path has to stick to the surfaces? For example, a string from one corner to the opposite corner but only allowed to run along the faces. I’ve seen people “unfold” the box and get something like sqrt((5 + 2)^2 + 8^2) depending on which faces you flatten. That gives a different number than the straight-through space diagonal. How do I know when I’m supposed to do the three-squares thing versus unfolding a net? And for points that aren’t opposite corners, is there a general rule for deciding whether I can use the add-three-squares trick, or do I need to project onto planes first? I’m probably overthinking this, but I’d love a sanity check!

I keep tripping over percentage increases when they happen more than once. If something goes up 10% and then another 10%, is that just a 20% total increase, or do I have to handle it differently? My brain wants to just add them, but I know that can be wrong.

What’s the simplest rule I can use in my head for repeated increases, without doing a bunch of steps? Example: start at 100, increase by 10%, then increase by 10% again – what’s the overall percentage increase?

Also, what’s the quick way to reverse it? Like if the final price is 115 after a 15% increase, how do I get back to the original fast without a calculator marathon?

I’m after a clean, practical shortcut here, not a lecture.

I keep tripping over the exponent in geometric sequences. I know one version is a_n = a1 * r^(n-1), but sometimes people start at a0 and then it’s a_n = a0 * r^n. My brain swaps them mid-problem and I lose a factor of r somewhere.

Simple example: start at 5 with ratio 2. For the 4th term, is it 5*2^3 or 5*2^4? Seems obvious until I switch to problems where they give two random terms. If a3 = 12 and a7 = 192, I get stuck on how to set the exponents so I don’t add or drop one power when solving for r or the first term.

Bonus confusion: if the ratio is negative (like start 8 with r = -1/2), I keep messing up which terms should be positive or negative.

Is there a dead-simple rule or mental trick so I don’t have to remember two different formulas? How do you keep the indexing straight without overthinking it?

How do you quickly tell, from y = -f(x-2)+3 vs y = f(-x+2)-3, which reflection it is and which way the shifts go-I keep messing up the inside-the-brackets stuff and left/right. Any help appreciated!

I’m practicing inverse proportion and I thought I had it: more workers means less time, so workers × time = constant for the same job. That feels super clean and satisfying. But I hit a word problem that made me second-guess everything.

Example: “6 machines make 480 parts in 5 hours. How long would 8 machines take to make 720 parts?” My brain says time is inversely proportional to machines… but also directly proportional to how many parts we need. So is this still an inverse proportion situation, or is it a combo of direct and inverse at the same time? I tried writing T ∝ 1/M at first, but then the output changed and that seems to break the ‘constant product’ idea.

I tried switching to rates: let r be parts per machine per hour, so M × r × T = total parts. That seems more sensible, but I’m not sure if I’m overcomplicating it or if that’s actually the right lens for “inverse proportion” problems. I also tried doing a quick proportion like 6/8 = T/? and immediately got tangled because the total parts weren’t the same-so I think I just did something illegal there.

What’s the quick way to detect when a situation is truly inverse proportion (like a pure hyperbola T = k/M) versus when I need to bring in another varying thing (like output or distance) and treat it differently? Also, as a follow-up: if I graphed T vs M for a set of targets (e.g., 480 parts vs 720 parts), is it basically a family of hyperbolas with different constants, or am I thinking about that wrong?

Why I’m stuck: I keep mixing up “inverse proportion” with “just take the reciprocal of the scaling factor,” and I’m not sure when I’m allowed to keep the product constant and when the ‘constant’ is secretly changing because something else changed. Any tips to keep my logic straight here?

I’m trying to get faster at mental multiplication for everyday stuff (like rough totals while shopping), but when I try two-digit-by-two-digit, my brain drops a gear and starts juggling jelly. For example, with 38 × 27 I try the distributive thing: (40 − 2)(30 − 3) = 40×30 − 40×3 − 2×30 + 2×3. I can start: 40×30 = 1200, then I subtract 120, and then I get wobbly-do I subtract 60 or 80 for the next part? And then I add 6, but by then I’ve lost track and my tens and ones are doing a conga line.

I also tried the “round then fix” idea: 38 × 27 = 38 × (30 − 3), which I rewrite as 38×30 minus 38×3. I can hold 38×30 in my head, but I keep mis-doing 38×3 (I say 90-something and then forget exactly what). Another attempt: halve/double to 19 × 54-felt clever, still got tangled.

What’s a reliable mental strategy to do this without dropping the cross-terms or place values? Is there a preferred order (like always do tens×tens, then tens×ones, etc.) or a neat way to keep a running total so I don’t mix up −60 and −80? If you can show how you’d step through something like 38 × 27 or 47 × 58 in your head, that’d be amazing.

Any help appreciated!

I’m stuck in surd-land and my brain keeps trying to “smush” things that apparently don’t want to be smushed. I know there’s a rule about multiplying under a square root that seems to work nicely, so I keep thinking adding should behave the same. But when I try something like √2 + √8 and squish it into √10… my calculator gives me a very judgey side-eye.

Here’s what I did (which I’m pretty sure is wrong):
– I tried √2 + √8 = √10.
– I also tried to split √12 into √6 + √6, which felt clever for about three seconds.
– And while rationalising denominators, I even did 1/(1 + √3) = 1/(1 + 3) = 1/4, which… yeah, probably illegal.

I’m confused because multiplying seems to let me combine things inside the radical, so why doesn’t adding work the same way? What are the actual do’s and don’ts here? If I have a simple example like √2 + √8, what’s the right kind of move I should be thinking about (if any)? And is there a simple reason why the “add inside the root” trick fails that I can keep in my head?

Thanks for any nudges-I keep looping on this and would love a way to stop making the same oops.

I’m trying to get better at quick estimates, especially with sums. I thought I had a neat trick: if I round some numbers up and some down, the errors should cancel as long as I have the same number going each way. Kind of like conservation of rounding energy… right? But it keeps not working and my brain is annoyed in a fun way.

Example: I wanted to estimate 2.01 + 2.40 + 2.60 + 2.80. I rounded to the nearest whole number: 2, 2, 3, 3, so my estimate is 10. I figured two went down and two went up, so the total should be pretty much spot on. But the actual sum is 9.81, which is off by 0.19. Why didn’t the “ups and downs” cancel here?

Is my assumption about cancellation just wrong? Is there a quick rule for when that logic works (or doesn’t), or a better way to estimate sums like this and predict the possible error? I tried pairing numbers (like thinking of 2.01 with 2.99-type partners) and also tried front-end estimation by adding the 2’s first and then the decimals separately, but I’m not sure if that’s the right approach.

Any help appreciated!

I’m trying to rotate a triangle on a coordinate grid by a given angle around a center that is not the origin. My plan is to translate so the center moves to the origin, rotate, then translate back. But when I plot the result, the image sometimes looks mirrored or it’s consistently shifted. I’m not sure if I’ve got the sign convention wrong for clockwise vs counterclockwise, or if I’m mixing up the order of the translations. I also worry I might be using degrees in one place and radians in another. Could someone clarify the correct sequence of steps, the sign to use for a clockwise angle, and any common pitfalls when the center is outside the shape? A small checklist would help me see where I’m going wrong. Any help appreciated!

I’m revising my fundamentals and trying to solve A = (b + c)/2 for b, but I keep tripping over the signs-can someone point out what I’m doing wrong? I multiplied both sides by 2 to get 2A = b + c, then wrote b = 2A + c; is that the right idea or am I missing a step?

I keep messing up Pythagoras in 3D-like when I tried to find the corner-to-corner stick inside a 1×2×2 shoebox for a DIY shelf (I messed this up before building a cardboard model in school!), I did sqrt(1+2+2)=sqrt(5) and also tried sqrt(1^2+2^2)+2 because it felt like walking along the edges in real life. Isn’t the 3D diagonal just “Pythagoras once and then add the third side,” or “square-root the sum of the lengths”-what am I missing?

I’m prepping for a test and my brain keeps trying to sneak marbles back into the bag when they’re not invited. Suppose I’m drawing two marbles from a bag without replacement. I know this is a “dependent events” situation, but I get tangled about how to think about it. Why exactly does the chance of the second color change just because of what happened first? Like, if the first marble is red, I get that the bag shrinks and the vibe changes… but is there a simple way to keep that straight in my head without accidentally using the independent-events mindset?

Also, here’s the part that really scrambles my cereal: if I draw the first marble and don’t look at it (I just set it aside like a mysterious marble of destiny), is the second draw still dependent? And follow-up: if the problem says I drew a red first but I put it back before drawing again, does that make the events independent again even though I “know” something about the first draw?

Could someone explain how to recognize these dependent situations at a glance and why the second probability actually shifts? Bonus points if you have a little mental trick for deciding when I should treat draws as dependent vs independent.

I’m okay expanding something like a number times a bracket, but as soon as a minus sign shows up I start second-guessing everything. For example, with 3(x + 4) – 2(x – 5), I keep forgetting whether that -2 should hit both terms inside the second bracket. Same with things like -(x – 7) or 1 – (x + 2) – do I flip both signs inside, or just the first one, or am I overthinking it? Is there a simple way to keep track of the signs so I don’t mess it up every other line?

Also, squared brackets melt my brain a bit. With (x + 3)^2, I know it’s not just x^2 + 9, but my brain really wants it to be. And when it’s mixed like (2x – 3)(x + 4), I lose track of which bits multiply and what the signs should be. Do you have any easy-to-remember tricks (like an area/box method or a quick mental checklist) to expand these without relying on me getting lucky? Bonus points if there’s a way to spot patterns like (a – b)(a + b) without memorizing a giant list.

I keep chanting BIDMAS like it’s a spell, but the middle bit is melting my brain. If D stands for Division and M for Multiplication, does that mean division “wins”? Or are they actually the same level and I should march left-to-right after brackets and indices? I get tangled on things like 12 ÷ 3 × 2 versus 12 ÷ (3 × 2). When there aren’t any extra brackets, how am I supposed to read it so I don’t invent a new number system by accident?

Bonus burr in my sock: expressions like 8/2(2+2). My brain happily does the brackets, then stares at the slash and the sneaky multiplication holding hands, and suddenly the whole thing feels like a logic seesaw. I’ve heard some people treat “implicit multiplication” (like 2(…)) as tighter than division, while others say multiplication and division are equals that go left-to-right. How do you decide what’s intended, and what’s a safe way to rewrite these so they’re unambiguous? Any tips or rules of thumb to keep my notebook from turning into a scribbly crime scene would be amazing.

I’m prepping for a test and I’m stuck: with class intervals of different widths, how do I set the bar heights so the areas tell the story-do I use frequency density or something else? I tried dividing each frequency by its class width, but I’m not sure that’s actually relevant.

I keep tripping over the sine rule when I’m solving for an angle. For example, I had a triangle with A = 30°, a = 7 (opposite A), and b = 10. Using the sine rule I got sin B = 10 * sin 30° / 7 ≈ 0.714…, so my calculator spits out B ≈ 45.6°. But then my teacher said B could also be 180° − 45.6° ≈ 134.4°, and now I’m second-guessing myself every time. I get especially lost when the diagram isn’t to scale, because I pick the wrong one and only realize later that the sides don’t make sense.

Is there a clean way to tell, just from the numbers, whether I should take the acute angle, the obtuse one, or consider both? Like a quick checklist for when there are 0, 1, or 2 possible triangles with the sine rule (that “ambiguous case” thing). And if both angles seem possible, how do you decide which one actually matches the given triangle without redrawing it ten times? I feel like I’m overthinking this!

I’m trying to mix a very particular purple for an art project, and my brain keeps doing somersaults over proportions. The guide says the color I like is 2 parts red to 5 parts blue. I’ve got exactly 350 ml of blue ready to go. I did this: red/blue = 2/5, so red = (2/5) * 350 = 140 ml. That feels right, but I’m second-guessing myself because sometimes I see people use parts of the total instead, like red = (2/7) * total. Are those two ways the same here, or am I mixing metaphors with my paint? Could someone explain which interpretation is correct and why? Also, follow-up: if I accidentally glug an extra 60 ml of red into the bucket (classic me), what would be the new red:blue ratio, and is there a clean proportional way to fix it by adding more blue instead of starting over? I keep tying myself in knots about whether to scale from the part I have or the total I want.