If you’re a normal student (or former student), you probably have asked yourself why you should know algebra. “I’m not a scientist, engineer, or doctor – what’s the point?” Well, a little algebra can help you make important financial decisions, as I found a little while ago.

I have a credit card that gives me 4% “cash back” when I pay for gasoline using my credit card. Since the bank’s not paying me for advertising, I’m not going to say which bank or credit card it is. ;-)

If you live in California or some other states, you may see most gas stations charge about 10 cents per gallon more for gas paid for by credit card than by cash. So the problem is – when does it become a better idea to pay with a credit card? [Of course, that’s assuming you pay it off at the end of the month – if you can’t do that, NEVER pay with a credit card unless you absolutely have no other choice but to buy the item now. Google “credit card interest compounded daily” for more details. Here’s one result I found: https://www.consumercredit.com/financial-education/financial-calculators/credit-card-interest-calculator But I digress. ]

The problem is simple enough : Just set up the inequality

(P + 0.10)*0.96 < P , where P is the cash price per gallon.

P + $0.10 < P/(0.96)

P +0.10 < 1.0417 P

$0.10 < 0.417 P

$2.40 < P

So if the cash price of the gas is more than $2.40 per gallon (as it does now), you save money with that credit card, even though you have to pay $2.50 when you use the credit card. Math is good!

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I never dreamed I’d have to write this, but since the same person has emailed me about this twice, here I go. I do not accept unsolicited content to post on my website. Any article, essay, blog, think piece, whatever you’d like to call it you send to me will not be read. In other words, as all the big publishers and production studios put it, “Unsolicited manuscripts will be returned unread,” assuming you’ve sent me a physical copy. If you’ve emailed me, I’ll just delete them without reading them. They’re not going up on any site I run.

Why am I being such a jerk, you ask? It’s simple – I don’t want to be sued by someone who says I promised them payment in the future for “free” content they sent me today. [See, for example, http://www.lawlawlandblog.com/2012/10/qa-why-are-unsolicited-submissions-policies-so-brutal-201210.html ].

That’s in addition, of course, to the obvious – I have no idea if your content is something with which I would agree, and finally, if your content’s so great, why not post it on your own website? You can get your own site literally for free (and the would-be contributor has her own site), and simply publicize your site. If I like your content, I’ll link to it (and maybe even tweet it), and you’ll get full credit for the article, since it will be on YOUR site.

I know that 99% of the people reading this have no intention of submitting anything for publication without my requesting it, but there’s always that obnoxious and/or oblivious 1%. Thanks for reading this!

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I'd be just as happy to help your student get a high test score, or with any academic or college admissions question.. I'm a full-service tutor. ;-) ]]>

2/7/18 – 2/12/18

2/17/18 – 2/19/18

4/6/18 – 4/8/18

6/19/18 - 6/25/18

7/17/18 – 7/22/18

10/14/18 – 10/28/18

11/22/18 – 11/25/18

12/21/18-12/26/18 (tentative)

12/28/18 – 1/2/19 (tentative)

I may also be in your area during those dates, and am willing to discuss meeting with you in person during my travel. For example, if you live in Los Angeles and I'll be there, I'll be glad to schedule a tutoring session while I'm in town.

Thanks,

John

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Here are a few notes I wrote for a student of mine who’s taking the SSAT Upper Level on Saturday, December 9. They might also help you study for the SAT, ACT, SSAT, ISEE, or any math test you may be taking.

The slope-intercept form of an equation is y = mx+b, where m is the slope, and b is the y-intercept (the value of y when x is zero). If you are presented with a formula such as 3y + 5x = 5, you have to rearrange the formula so you get y = mx+b. In this case, it would be y = (-5/3)x + 5/3.

If a line is perpendicular to another line, the slope of the perpendicular line is the negative reciprocal of the first line. For example, if the first line has a slope of 2, the perpendicular line has a slope of –½ . Just flip the fraction (or put one over the slope if the slope is an integer), and then multiply it by -1. So the slope of a line perpendicular to a line with slope 3 is -1/3 and the slope of a line perpendicular to a line with slope 33/44 is -44/33. Notice that b, the y-intercept, doesn’t matter when you’re finding the slope of a perpendicular line – all b would affect is WHERE the perpendicular line intersects the first line, not the angle at which it intersects the line.

The imaginary number (or imaginary unit),

See, for example, this page from the publishers of the “SSAT & ISEE for Dummies” book we’ve been using:

http://www.dummies.com/education/math/algebra/how-to-multiply-binomials-using-the-foil-method/

For an example of how to do the same with complex numbers, see the top of this page:

http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U16_L4_T2_text_final.html

Remember – the imaginary unit,

As we saw in the problem we did tonight, A with a bar over it like this : Ā means the complement of Set A, which means everything in the “universal set” or “universe” of all things in all the sets we are considering that are not in Set A. So if the universal set, U, contains the numbers 1 through 10, inclusive, and A is the subset of U containing 2,4, 6, and 8, Ā is 1,3,5,7,9, and 10. This could also be represented using brackets to hold the set – that is Ā = {1,3,5,7,9, 10}.

You should also know the terms “union” and “intersection”, and their symbols, U and ᴒ (this is the closest I could find to the symbol, which looks like an upside-down capital U). So {1,2,3} U (4,5,6} stands for the union (joining) of those sets into one set {1,2,3,4,5,6}. The union of two or more sets just makes a bigger set containing all its members.

The intersection of two or more sets includes only the members common to all the sets involved. So {1,2,3} ᴒ {3,5,7} would be the intersection of those two sets. Since they only have one common member, that set is {3}. The intersection of the set of all prime numbers, all even numbers, and all numbers less than 10, would also have only one member, the number 2. Questions could be asked that way, in words rather than set notation, perhaps using sets with infinite numbers of members, such as prime numbers, even numbers, numbers less than 10, etc.

The area of a rectangle is equal to the length times the width.

The area of a square is the side length times itself. That’s why a number multiplied by itself, or taken to the second power, is called that number “squared.”

The area of a circle is , where r is the radius of the circle, and is a constant that is approximately 3.14.

For more information on area, see the following page:

http://www.mathsisfun.com/area.html

You should know the volume of a cube, cylinder, cone, rectangular solid (i.e., a box), and a sphere. Bonus points if you know the formula for a pyramid and a tetrahedron. Here’s a page with those formulas and more.

http://onlinemschool.com/math/formula/volume/

You should know that the surface area of a cube is 6(e^2), where e is the edge length. Be prepared to find the surface area of a cube when given the volume – e.g., if the volume of a cube is 27, the edge length is the “cube root” (remember, the 3rd power of any number is that number “cubed,” get it?), which is 3. Since there are 6 square sides, or “faces,” the surface area is 6*(3^2) , which is 6*9 = 54.

You may also need to find the surface area of a cylinder, which is shaped like a can. So it has two circles on the top, and a rectangle twisted around the middle. So find the area of the circle on the top and bottom (they’re the same), using the radius or diameter or circumference of the circle (depends on what they give you), then multiply the circumference of the circle times the height of the cylinder. Then add the areas of the top, the bottom, and the side.

See, for example:

https://www.google.com/search?q=surface+area+of+a+right+circular+cylinder&pws=0&gl=us&gws_rd=cr

and

http://www.learnalberta.ca/content/memg/Division03/Cylinder/CylinderSA/index.html

The relationship between the speed at which you travel and the distance travel is D=RT, where D is the total distance traveled, R is the rate (speed) at which you travel, and T is the time you traveled. So if you traveled 50 miles per hour for two hours, the distance you traveled is 50 miles/hour * 2 hours = 100 miles. For some more complex distance -rate-time problems, see:

https://www.gcflearnfree.org/algebra-topics/distance-word-problems/1/

http://www.purplemath.com/modules/distance.htm

http://www.analyzemath.com/math_problems/rate_time_dist_problems.html

These are similar to distance problems, in that the formula for work done is W = RT, where W is the amount of work done, R is the rate at which the work is done, and T is the time. So we’re just substituting work for distance in the distance formula.

HOWEVER, work problems on most tests I’ve seen are usually different from distance problems because two machines or people can work on the same job, shortening the time required to do the job. So you’re likely to run across a test problem such as “Printer A can print a huge print job in 3 hours, and Printer B can do the same job in 5 hours. How long would it take Printers A and B, working together at the same rate, to do the job?” This problem requires multiple applications of the work formula to solve.

First, we solve for RA, the rate at which Printer A can do the job by itself. RA = 1 job/3 hours, when we divide R by T. RB = 1 job/5 hours, using the same logic. Therefore, when both printers work together, we get 1 job = (RA+RB) TA+B, where TA+B is the time it takes Printers A and B, working together at the same individual rates, to do the job. So TA+B = 1/ (RA+RB), which is 1/(1/3 +1/5) .

So we have to find a common denominator for 1/3 and 1/5.The best common denominator to use is the lowest common multiple of both denominators. If you don’t know how to find the lowest common multiple of two numbers, see here: http://www.math.com/school/subject1/lessons/S1U3L3DP.html

Since there are no common factors of 3 and 5, we just multiply 3 and 5. We take 1/3 * 5/5 to get 5/15 for RA

Hope this helps!

Sample SSAT Test:

http://ssatprep.com/free-practice/

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All page references are to

In addition to knowing what makes a complete sentence, and how to join two clauses with a FANBOYS (For And Nor But Or Yet So) conjunction please review the following.

Wordiness (aka “What Would George Carlin Do?”): p. 247

Parallelism (aka “What the heck is parallelism?”, “Why is it important?” and “How do I do it?”): p. 248

Modifier Placement (aka “Misplaced Modifiers” – also explains what modifiers are and why they should be properly placed): p. 249.

Logical Comparisons (aka “How to Avoid Letting a Smart-Aleck Make You Sound Stupid”) pp.250-251.

Writing Test: Verb Tense and Use and Irregular Verb Forms (aka “Remember 7th Grade English class?): pp. 252-253.

American idiom – Common Idiomatic Expressions ("We Say It This Way Because We Say It This Way. Stop Asking Questions!" – pp. 259 – 260.

Also punctuation – Commas, Colons, Semicolons, Dashes, and Apostrophes ("More Stuff You Should Have Learned by Now") – pp. 261-267.

Finally, study the vocabulary for the reading and writing tests – found at pp. 161 – 236. You can also print out flash cards at barronsbooks.com/tp/sat .

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The sum of the internal angles for a polygon with

The sum of the external angles, taken one per vertex (corner) for any polygon, no matter how many sides it has, is 360 degrees. For two external angles per vertex, multiply 360 by 2 to get 720; for 3, multiply 360 by 3 to get 1080 degrees.

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Here’s a quick list of formulas and other math facts you should know for tomorrow’s SAT. You will have to memorize them; they will not be provided in the “Math Facts” box at the beginning of the math sections.

The “difference of squares”: (a+b)(a-b) = a2 – b2

Other binomials:

(a+b) = a2 +2ab + b2

(a-b) = a2 - 2ab + b2

A = d2/2. This comes from the fact that a square’s diagonal splits it into two 45-45-90 triangles, which have the side ratio of 1-1-√2, so the side length, s, is equal to d/√2, and the area, which is s2, is then equal to (d/√2)2, which is d2/2.

Know that a square’s diagonals bisect, are congruent (equal), and are perpendicular, to each other.

- Angles that are diagonal to each other are equal.
- Consecutive angles (ones that are next to each other) are supplementary (they add up to 180 degrees).
- If one angle is a 90-degree (right) angle, then they all are.
- The diagonals bisect each other.
- Each diagonal bisects the parallelogram into two congruent triangles.

Properties of transversals (two pairs parallel lines that cross each other).

Notice that parallelograms are formed by transversals (two pairs of parallel lines).

Notice that angle 1 = angle 8 = angle 2 =angle 5 and angle 4=3=7=6. If there were a line parallel to line t, the corresponding angles formed with line l and m would also be equal (congruent) to the angles line t formed with lines l and m (that is, all the acute angles are congruent, and all the obtuse angles are congruent). See the first diagram in this section. Obviously, if they were all right angles, all the angles would be congruent, since they’d all be 90 degrees.

**Area of a Trapezoid:**

Here, if you know the side length is s, then you know you can split the triangle into two 30-60-90 triangles with side lengths s/2, s, and s√3. The s√3. side is the altitude of the triangle that splits it in two. So the area of one 30-60-90 triangle is 1/2 * s√3. * s/2, and the area of the whole equilateral triangle is (s*s√3)/4, which is (s2√3)/4. You can memorize this formula, or just know how to derive it by using an altitude to split the triangle in two.

I hope these formulas and facts help. Please read my other blog entries for other tips. Good luck on the SAT

]]>I hope these formulas and facts help. Please read my other blog entries for other tips. Good luck on the SAT