I am really having a hard time understanding translating word expressions into equations, and wondering if you can help?

Example:
1. the sum of 16 times a number and the number less another number times 3

2. a number less the sum of another number and 13

Workin’ these problems is almost like translating another language. You gotta know exactly what each word translates to a mathematical expression. Here’s a list of how they break down:

+ = “the sum of x and y (you’d put the + where the “and” goes here: x + y)

− = “less” or “minus” (note that if it says something like “a less b” then it’s a − b but if it says “a less than b” it’s b − a)

× = “times” or “the product of a and b” (a times b = a × b)

÷ = “divided by” (a divided by b = a ÷ b)

a,b,c or x,y,z = These are variables, so if it say “a number” that’s one of these variables. You usually wanna start with the first one in a list. Then if it says “another number” you pick the next one. If it said “a third number” you’d pick the last one. I don’t think it’d ever go above three numbers. (a number = a, another number = b, a third number = c)

Where it can get tricky is like when you start gettin’ a bunch of problems all mashed together. So like, take the first one that Trish posted.
“the sum of 16 times a number and the number less another number times 3″

You wanna write this stuff down on paper:

So, I put red parenthesis around the different parts of the problem. Is says the sum of “this” and “this” so you want to make sure to separate the two. Then you can translate things directly:

(16 × a) + (a − b × 3)

Then you can simplify it down:

16a + (a − 3b)

16a + a − 3b

17a − 3b

In this problem, the parenthesis didn’t really do anything. But it’s important to keep things separated in case it does matter. You wanna get into the habit of looking for the way problems are broken apart. Like, if it said “the product of “…” and “…” then the parenthesis would be crucial ’cause you gotta make sure to do any addition inside the parenthesis before you do the multiplication. If you check out the next problem, parenthesis play a bigger part.

So, we got “A number less the sum of another number and 13.” Translating it directly, we get a − b + 13. But since it says “less the SUM” you gotta use the parenthesis to separate the first part, “a number” and the second part, “the sum of another number and 13.” So it gotta look like this:

a − (b + 13)

Now when we simplify it, we distribute the negative:

a − b − 13

As a side note, make sure to read the directions of a problem carefully. If they just want you to write the equation out as a mathematical expression, the answer’d be a − (b + 13). If they want you to simplify, it’d be a − b − 13. If they want you to solve for a number, like a, then the answer would be a = 13 + b. A lot of the time, simply missin’ things like that in the question can make a problem wrong, even though you actually knew what you was doin’.

Good luck, all.

For more information about the GED test and GED test preparation, visit the GED Academy at http://www.passGED.com.

I been hearing about this thing lately called dyscalculia. I ain’t never heard of that before, but I have heard ’bout dyslexia. So, I figured the two were linked. I looked it up, and it turns out they are. Discalculia’s kinda like dyslexia with numbers. Only it’s more than that. It effects people’s sense of time and space and all that. Check out this list of symptoms.

Anyway, it kinda helped me understand people a little better. Like, when I add up numbers, I just get it, you know? But if I think about it like dyslexia, that’s somethin’ I can understand. Sometimes I just don’t get words. It’s like they all a jumble, and I gotta slow down and really pay attention. But there’s people out there who can just scan a page real quick and tell you everything that be on it. Maybe those same people can’t get numbers like I can, right?

So, maybe some people got discalculia without knowing it. It’s not well known, like dyslexia. Maybe someone famous gotta have it first before the public notices it. For now, there’s a cool site called Dyscalculia Forum that’s got a lot of info and other people who’ve got dyscalculia. They help each other out and offer up solutions they’ve found that helps them remember numbers.

Mostly, it seems like if someone’s got problems in one area, they probably are pretty good at somethin’ else. So, if you’re having problems with numbers, you gotta think about somethin’ else you’re good at, like words, art, or music. For instance, there’s ten numbers total, right? Maybe you can assign a color to each number. Like this:

0123456789

So, 0 is black ’cause it’s nothin’ so it’s not a color too, right? Then I started with pink, red, orange, yellow, green, teal, blue, purple, violet. Someone who’s real artistic might be able to remember the order of colors easier than the order of numbers. And maybe they can remember mental multiplication easier with colors. So instead of 7×8=56, they might think, blue x purple = green-teal.

I don’t think in colors myself, and that’s a pretty wild example, but it shows how you might get started thinkin’ about different ways to understand numbers. Here’s some other ways to think about numbers differently:

• Read problems aloud and talk through the answers (sometimes hearing yourself problem solve is helpful).
• When learning a new concept, make sure you understand it well enough to teach it back before moving on.
• Try to visualize the Math problem. If the problem is about a house, draw the house, then add in the dimensions as the problem goes along.
• Practice estimating as a way to solve problems.
• Don’t be afraid to count on your fingers.
• Use scratch paper! You may remember things better by writing them down and working through the problem on paper and not in your head.
• Try using colored pencils for emphasis or to differentiate problems.
• Memorize Math facts to music or a beat (Mary had a little lamb, etc.).
• If you are doing a “non story problem” type of Math problem, make up a story for it. If you can relate the problem to real life, it may be easier to solve.

When it comes down to it, though, everyone’s different, so you gotta figure out a method that works for you. Check out these 8 different types of intelligences that Dr. Howard Gardner discovered. Maybe you can figure out which one fits with you and come up with your own strategy for understanding Math better.

1. Linguistic and verbal intelligence: good with words
2. Logical intelligence: good with math and logic
3. Spatial intelligence: good with pictures
4. Body/movement intelligence: good with activities
5. Musical intelligence: good with rhythm
6. Interpersonal intelligence: good with communication
7. Intrapersonal intelligence: good with analyzing things
8. Naturalist intelligence: good with understanding natural world

For more information about the GED test and GED test preparation, visit the GED Academy at  http://www.passGED.com.

Hi Curtis,

I need some help with percent and ratio word problems. Unfortunately your previous explanations regarding word problems have been too complicated. Perhaps you could give more information on the basics, the formulas? I know I am not completely understanding these formulas. My knowledge in math is only the basic concepts, and I do not understand algebra yet.

While percents seem simple enough; I become lost when I try to solve word problems with them. I have been using the triangle method to work with percent problems. [The method shown in the GED book.]

1- Multiply when the problem gives you the whole and the percent.

2- Divide when the problem gives you the part and the percent.

3- Divide when the problem gives you the whole and the part.

However, I am still finding word problems with percents and ratios very confusing, so I know I am definitely not understanding the formula. Ratios especially – the whole idea of cross multiplying sounds good, but when I do this I become lost as I attempt to finish the problem. I hope you can help me begin to make sense of these areas.

Thanks, Jen

Hey, Jen. The problem with percent and ratio word problems is, you gotta really think through what information they give you an’ how it relates to the problem they want you to solve. What are you actually tryin’ to find? How can you get it? It’s knowing when to use the different rules and formula’s that’s confusin’.

Like the triangle rules you said you use:

1- Multiply when the problem gives you the whole and the percent.

2- Divide when the problem gives you the part and the percent.

3- Divide when the problem gives you the whole and the part.

To use ‘em, you gotta know what ‘percent’ ‘whole’ an’ ‘part’ they mean. Here’s an example.

40 of the students in a class of 200 got B’s on their test. 10 got A’s, 2 got F’s, and 20 got D’s. What percentage of students got C’s?

What’s the whole, part, and percent?

The whole is the whole class: 200.

The percent is the percent of students got C’s. That’s what you’re looking for.

The “part” is the number of students that got C’s.

See, a “percent” is the fraction (or ratio) of the part to the whole. That’s all… it’s just a fraction, with the top divided by the bottom. So, 1/2 = 50% (1 divided by 2 = .5 = 50%) and 3/4 = 75% (3 divided by 4 = .75 = 75%). The part divided by the whole = the percent.

Part ÷ Whole = Percent

So, in a sense, a percent is the same thing as a fraction. 1/8 of a pizza is the part (1 piece) over the whole (8 pieces make a whole pizza). That same slice of pizza is 12.5% of the pizza, because 1 divided by 8 = .125 (12.5%).

All three of the “rules” come from the equation: Part ÷ Whole = Percent, just written in different ways.

1- Multiply when the problem gives you the whole and the percent. (Part = Percent x Whole)

2- Divide when the problem gives you the part and the percent. (Whole = Part ÷ Percent)

3- Divide when the problem gives you the whole and the part. (Percent = Part ÷ Whole)

All these formulas say the same thing… it’s just moving the three pieces of the formula around. So really, all you need to do is figure out what the question’s askin’, and find the other two numbers to plug into the formula.

Back to the word problem. It’s asking for a percent:

Percent = Part ÷ Whole

The “whole” is the whole class, 200 students.

Percent = Part ÷ 200

The “part” is trickier, cuz the word problem is really in 2 parts. The “part” is the number of students that got c’s. So, you need to subtract all the other students who got other grades from 200.

200 – 40 – 10 – 2 – 20 = 128

That’s the number to plug into the formula:

Percent = 128 (number of students with C’s) ÷ 200 (number of students in the class)

128 ÷ 200 = .64 = 64%

64% of students got C’s.

Okay, let’s try a ratio. A ratio is also sort of like a fraction, but it’s not necessarily the ratio of “part” to “whole.” A fraction is a ratio of “part” to “whole,” but a ratio is a ratio of anything to anything. In a speed problem, it can be a ratio of “miles” to “hours.” It’s anything that has a regular relationship. Okay, let’s try a ratio problem, and we’ll try an’ walk thru cross-multiplying.

A man works 8 hours per day in a factory, and he makes $20 per hour. In a 5-day workweek, how much does the man make?

Now, you might figure this out without even using a ratio. But it is a ratio… any time you see “per,” you can use a ratio. You need to figure out what’s equivalent to what. We’ve got some ratios:

8 hours : 1 day

That’s the same as 8 hours per day.

20 dollars : 1 hour

That’s the same as $20 per hour.

Now, if I want to find out how many dollars per day, I need to figure out the ratio of dollars to 8 hours. A ratio like this (something per something else) gives two things that are equivalent. 8 hours = 1 day, so a ratio of dollars per 8 hours will give us dollars per day.

20 dollars : 1 hour

x dollars : 8 hours

Now, I’ve got two ratios, with the same units, and they’ll be equal to each other. That gives me an equation:

20/1 = x/8

Now’s the time to cross multiply. Basically, cross multiplication is some simple algebra. It’s just a shorthand way to remember how to solve this kind of problem. To move numbers from one side of the equation to the other, you do the opposite operation. If one side is divided by a number (like 1), you can move that number to the other side of the equation by multiplying both sides by 1.

(20/1) x 1 = (x/8) x 1

The two ones on the left cancel each other out, and on the right you end up with x times 1 (or just x) over 8:

20 = x/8

Now, do the same thing with the 8:

20 x 8 = (x/8) x 8

The two 8′s cancel each other out, and 20 x 8 = 160:

160 = x

He makes $160 in a day.

160 dollars : 1 day

Now, how much does he make in 5 days?

y dollars : 5 days

It’s the exact same sort of problem. You can make an equation:

160/1 = y/5

Using the shorthand version of the math we did before, this is the same as 160 x 5 = 1 x y (cross multiplying)

160 x 5 = 1 x y

800 = y

That’s the answer: he makes $800 in a five-day workweek.

So, what you need to find out from a ratio problem is what ratio equals what other ratio. Then, you can cross-multiply to solve.

Hope this helps! If a practice problem is giving you problems, send it to me an’ I’ll walk thru it.

For more information about the GED test and GED test preparation, visit The GED Academy at http://www.passGED.com.

Hey Curtis ,

Thanks for your prompt reply man . This is one of the problems I struggled with involving distance :

- A man started walking at 2 mph, while a woman 2 miles behind him began walking at the same time at a rate of 4 mph, and in the same direction. Just then, the man’s dog left him and ran toward the woman. Upon reaching her, it instantly turned around and ran back toward thr man. And so, the dog continued to run back and forth between them, at a constant rate of 5 mph, until the woman finally overtook the man. How far did the dog run?

**** Go enjoy figuring it out and let me know how to do it man !

Zaher

In the GED test, on the page with all the formulas, you got one for distance:

distance = rate x time

So, what’s it mean? Distance is how far you gonna go, rate is how fast you goin’, and time is, well, time. So, how far you go is how fast you go times how long you’re travelin’. In other words, if you get in your car and start drivin’ down the highway, the faster you go, and the longer time you drive, the farther you’ll get. Makes sense, right? If your speed (rate) goes up, the distance you travel goes up. If your time (how long you drive) goes up, the distance you travel goes up.

How does it get applied to the problem? This problem is tough, not because the math is tough, but because it’s tough to figure out how to use the math to solve it. You gotta take it one step at a time and figgure out what’s really goin on here.

Okay, we got a guy walkin’. Same idea as driving, how far = how fast x time. And, we got a woman walking, to. So that’s complicatin’ it up. We got to deal with 2 people walkin’ at once. And a dog, runnin’ back an fo’! Okay, problem with this problem is it’s hard to see. What’s the real question? How far’s the dog go? How do I figure that out?

Hey, this is harder than anythin’ on the GED. So, let’s stretch our minds. To get my mind around it, sketchin’ a picture sometimes helps. Scuse my bad drawin’.

So, the real point is, how far does the dog go? We want to know distance (how far), which is rate times time:

dog distance = rate x time

dog distance = 5 mph x time

Okay, to answer the question, we run into another question… how long does the dog run? He runs from the time they start walkin’ until the two people meet. Now, we gotta answer a second question to answer the first. How long does it take the woman to get to the guy?

We’re usin’ the same formula, an’ tryin’ to find the time it takes for the woman to catch up.

distance = rate x catchup time

The distance they start out is 2 miles:

2 miles = rate x catchup time

What’s the rate? The man is walking at 2 mph, and the woman is walking at 4 mph. She’s catching up to him, but how fast? For every 4 mph she walks, you’ve gotta subtract the 2 mph he’s walked in the same direction during that same time. So, she’s gaining 2 mph on him.

2 miles = (4 mph – 2 mph) x catchup time

2 miles = 2 mph x catchup time

Now, it’s just math. divide each side by 2, and you get:

1 hour = catchup time

How do you know the units are hours? It’s 1 hour, because we’re talkin’ miles and miles per hour. So, one question solved! Now we take the answer and go back to our original question:

dog distance = 5 mph x 1 hour

dog distance = 5 miles

Whew! Dat dog goin’ doggone fast!

For more information about the GED test and GED test preparation, visit The GED Academy at http://www.passGED.com

So, let’s try walkin’ thru one.

Jerry wants to buy twelve pizzas. The pizza place has a discount special, where you buy 2  pizzas and get the third 1/2 off. If P is the price of a pizza, which formula shows the price of twelve pizzas?

A) 12P/2 + 2P

B) 1/2 x 12P

C) 12P – .5P

D) 8P + .5(4P)

Okay, so what do these formulas really mean? Let’s work it from the formulas, to see if they match up with the question.  P is the price of pizzas. So, the first formula means:

12P/2 + 2P means 12 pizzas, divided by 2, plus 2 pizzas. So, twelve pizzas are half off (that’s the divided by 2), and 2 pizzas are full price. That ain’t right. He’s buyin’ twelve pizzas, not 14, and he’s not gonna get half off of 12 of them.

Let’s try the next one. 1/2 x 12P means 1/2 off of 12 pizzas. Well, that’s not right. Only every third pizza is half off.

How ’bout the next one?  12P – .5P means twelve pizzas minus the price of half a pizza. That’s only one pizza being half off. That’s not right. He oughtta have more than that off his total.

Only one left.  8P + .5(4P) means 8 pizzas plus half of 4 pizzas. Point-five means half just like a fraction. So, is that right? If it’s buy 2 get 1 half off, there should be twice as many full price pizzas as half price pizzas. So there are. 8 is twice 4. So for every 2 full-price pizza, he pays half for one other pizza. And there should be 12 pizzas all together. And there are. 8 pizzas and 4 half-price pizzas, that’s 12. There’s your answer.

Let me know if you’ve got any questions.

For more information about the GED test and GED test preparation, visit The GED Academy at http://www.passGED.com or call 1-888-880-2164.

Take a deep breath! Here’s a breakdown of the formulas. Remember, they’re in the GED test booklet to help you out….

So, what’s it mean? Area is just the amount of space on a surface. That’s all. For a square or rectangle, it’s just the length of two adjoining sides multiplied by each other. So, if you’ve got a 6×4 room, you need 6×4 square feet of carpet…24 square feet.

A parallelogram is just a four-sided figure where the opposite sides are parallel…slanting the same way. It’s like a rectangle, but instead of measuring the length of the sides, you measure 1 flat side, and then the height, straight up. Multiply them together to get area.

A triangle is easy…it’s half a parallelogram, so you get the area by multiplying 1/2 times the base (bottom) times the height (straight up).

A trapezoid is a 4-sided figure with 2 parallel sides. So, the sides that run parallel to each other are different lengths. Add them together and divide by two to get an average…then multiply by the height (straight up).

A circle is a little different… you need to use “pi.” That’s that funny-looking figure…just think of it as 3.14. To get the area (flat surface) of a circle, multiply 3.15 times the radius squared. The radius is the distance from the center of the circle to the edge, and squared just means multiply it by itself. So, if the circle is 4″ across, the radius is 2″, and the area is 3.14 x 2 x 2, or just over 12. (Approximating can be very helpful!)

Next up is perimeter and circumference…

Perimeter is just how long the outside lines of a shape are…so, how much fence you’d need to put around a pasture, or how much framing you’d need to frame a picture. On the square, rectangle, or triangle, or any figure with straight sides, it’s just the length of all the sides added together. Easy.

On a circle, you’ve got to use pi again…so it’s 3.14 x diameter—that’s the length across the center of a circle. For a 5″ across circle, the circumference is just over 15.”

Next is volume…

Volume is like area, except it’s three dimensional. It’s all the space inside something. For area, you multiplied one side times another, right? Well, for volume, you’re just adding a third side… so for a cube or a rectangular solid (like a box) you multiply length x width x height.

A cylinder is like a circle that’s got height. So, for the cylinder, you find the area of the circle at the bottom (pi times radius square, like we did for area), and then multiply it by the height of the cylinder.

Cones and pyramids have a circle or a square at the bottom, and then they come to a point instead of having a similar shape at the other side. So, they’re smaller in volume than a cylinder or rectangular solid. Turns out, they’re exactly 1/3 the volume. So, just find the volume of a cylinder or rectangular solid with the same size end, and divide that number by three. That’s all the formulas mean.

Now the next one…

COORDINATE GEOMETRY
	distance between points = ; (x1, y1) and (x2, y2) are two points in a plane.Slope of a line = ; (x1, y1) and (x2, y2) are two points on the line.

Sounds confusing! But it’s not so bad, really. Points on a graph are shown by an x and y number, like this: (2, 3) The x number is the first number, and the y number is the second number. The numbers tell you where to find the points on the graph. Well, to find the distance between two points, you basically make a right triangle on the graph, by connecting the points. Then, you can use the Pythagorean formula to find the distance…

The Pythagorean idea is that, in a right triangle, the length of one (short) side squared + the length of the other (short) side squared = the length of the long side squared. That’s what you’re doing here. The distance between the x’s is the length of one short side, and the distance between the y’s is the length of the other short side.
Find the distances, square them, add them together. Then, find the square root…and you’ve got the distance between the two points.

The slope…that sounds confusing, too. But basically, it’s RISE over RUN….that is, how far it is up and down from one point on a line to another, over how far it is across between the same two points.

The next one is the Pythagorean relationship…exactly what I was just talking about. a and b are the two short sides of a right triangle (legs), and c is the long side (hypotenuse). You’ll use this formula whenever you know the lengths of two sides of a right triangle and want to know the third. This one’s a good one to know!

MEASURES OF CENTRAL TENDENCY
	mean = ; where the x’s are the values for which a mean is desired, and n is the total number of values for x.
	median = the middle value of an odd number of ordered scores, and halfway between the two middle values of an even number of ordered scores

“Mean” is what we usually think of as an “average.” In plain English, add up all the numbers you’ve got, and divide by however many numbers you added together.

“Median” is just the middle number, if you put a group of numbers in order from smallest to largest (that’s what it means by ‘ordered scores’). If there are an even amount of numbers, there won’t be a middle number…so you use the number halfway between the two middle numbers.

The last items are simple interest….so if the GED asks you about interest on a loan, you’ll calculate interest by multiplying the principal (amount borrowed) by the interest rate and multiplying that by the amount of time of the loan (usually in years).

Distance = rate x time is about how far you can go, how fast. So, if you’re traveling at 30 miles per hour for 6 hours, you’ll go 30 x 6 miles, or 180 miles.

Total cost = (number of units ) x (price per unit) is something you do every day…
if you’re buying 5 bananas, and bananas cost 20 cents each, how much is the total cost? It’s the number of bananas times the price per banana…or 5 x 20 cents, or 100 cents, or $1.

Don’t let the formulas confuse you! Most of it is pretty straightforward…

Good luck! And don’t panic!