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OK, so you have 40 boys and 30 girls. That’s easy enough to calculate because you’re given a part:whole ratio (boys to total students) and you already know the total number of students is 70.

Be careful about the second ratio, though, because it’s a part:part ratio! If the ratio of older to younger is 2:3, then the ratio of older to ALL STUDENTS is 2:5. (To figure this out, imagine that you only have 2 older kids and 3 younger kids). So you have 2/5*70 = 28 kids 10 or older, and 70 – 28 = 42 kids younger than 10.

To finally answer the question, we know 2/3 of the girls, or 20 girls, are less than 10. That means the other 42 – 20 = 22 kids younger than 10 must be boys.

If there are 40 boys, and 22 of them are younger than 10, then the other 18 must be 10 or older.

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OK, so when you have a regular n-gon, you can figure out each angle in it using this formula: [(n-2)180]/n. In this case, 7*180/9 = 140, so we know each angle in the polygon is 140°.

I couldn’t draw this quickly on the computer I’m on, so I found a good n-gon picture to mark up from Wikipedia. 🙂 Please forgive the kinda sloppy graphics below once I start marking up the figure.

By László Németh – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=27295121

Now here’s what we care about:

We can get at the measure of ACE by considering that triangles ABC and CDE are isoscelese (because this is a regular polygon), and that angles B and D are 140°. Therefore, the small angles in those triangles must all be 20° (because triangle angles always add to 180°).

We know angle BCD is also 140°, and we’re taking 20° from it on either side. Therefore, the measure of angle ACE is 140° – 20° – 20° = 100°.

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The midpoint formula tells you that the a segment with endpoints (a, b) and (c, d) will have a midpoint at ((c)/2,(d)/2)

So we know that (+ 9)/2 = 6 and (5 + y)/2 = 3. We can solve those!

(+ 9)/2 = 6
+ 9 = 12
x = 3

(5 + y)/2 = 3
5 + y = 6
y = 1

Therefore, x + y = 4.

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When a system of linear equations has no solution, that means you have parallel lines, which means the lines have the same slope. So put both equations into slope-intercept form (y = mx + b) first:

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In order for those lines to be parallel, their slopes must be equal, which means 2/5 = -4/k. That means k must be equal to –10.

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Just do those things and simplify:

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image

Looks like the volume of the second cylinder is half the volume of the first! 🙂

The shortcut here is to recognize that the r value gets squared but the h value doesn’t, so you can say the volume is divided by 4 then multiplied by 2.

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60° is 1/6 of the circle (which has 360° of arc in total), so the length of the minor arc will be 1/6 of the circumference. 1/6 of 12π is 2π.

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Huge shortcut here if you just know that for a parabola in standard ax^2 + bx + c form, the x-coordinate of the vertex will be at –b/(2a). In this case, that means it’s at –3/(2(–6)) = 3/12 = ¼.

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The way to think about this (for me, anyway) begins with understanding that 2[something] + 3 = 8x – 1, and our job is to figure out what that something is.

Since this question gives us answer choices, all we really need to do is try each one as the something to see what works. Since they’re not in numerical order, start with A.

2(4x – 2) + 3
8x – 4 + 3
8x – 1

Oooh, look at that! we’re already done! 🙂

(Also, for those worrying along at home, note that 5 answer choices means this isn’t an SAT question. Could be SAT Subject Test, though.)

If you didn’t have answer choices, you’d still start in the same place, but then think through logically. Thought process:

2[something] + 3 = 8x – 1

Hmm..there’s gotta be a 4x in the something otherwise i’ll never get 8x. But if it’s just 4x in there then I’ll end up with 8x + 3, not 8x – 1. How do i take 4 away? By adding a –2 inside the something!

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When you have exponents with the same base and you divide them, you subtract the exponents.

So from what we’re given, we know that 2x – (xy) = 1. We can solve that for y.

2x – (xy) = 1
2x – x + y = 1
x + y = 1
y = 1 – x

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Start by drawing it!

Note that OC = 5 and OD = 5 because both of those are also radii. Note also that because chord CD is perpendicular to OB, it’s bisected by OB. In other words, it’s split into 2 segments each measuring 4.

 

Things are really coming together! Because we know our Pythagorean triples when we see them, we know that we’ve got 3-4-5 triangles here. Therefore, the sine of angle OCD is 3/5.

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One of the two things being multiplied must be zero, so either x – 3 = 0 or x + 7 = 0.

If x + 7 = 0, then x = –7. In that case, we can substitute into x – 3:

x – 3
= –7 – 3
= –10

So x – 3 can equal 0 or –10.

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Draw!

You know point C must be on the x-axis, and that AB = BC. Since B is on the y-axis, it turns out all we need to do here is reflect point A across the y-axis. Point C must be at (8, 0), so x = 8.

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The correct answer is A. Subtract c from both sides of the inequality and you’re left with 8 > 4. That’s always true!

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In 200 ml of 30% saline, there’s (0.3)(200) = 60 ml of saline.

60 ml is 40% of what?

60 = (0.4)x
150 = x

So the saline will be 40% when enough water evaporates that there’s 150 ml of solution. Therefore, 50 ml of the original 200 ml must evaporate.

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Let’s work backwards, and use hours instead of distance for ease (assume the plane travels at a constant speed for our purposes).

Let’s say when she woke up, she had 1 hour left in her flight. That’s half of the time she was sleeping so she must’ve slept for 2 hours.

She first fell asleep halfway through the flight, so she was awake for the first 3 hours, then slept for 2, then was awake for the last 1.

She was asleep for 2/6 = 1/3 of the trip.

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