Math: solving percents with cross-multiplication

As a math tutor, you encounter this topic often – especially with students in vocational training.

Many ways exist to tackle percents.  However, the advantage of the cross multiplication method is its usefulness in virtually any situation involving them. If you missed my blog entry introducing the cross multiplication method, find it here.

Example 1: What is 15% of 390?

The key with percents is to realize that, for instance, 15% means 15 over 100. That is,

(1)   \begin{equation*} 15\%=\frac{15}{100}\end{equation*}

We can now incorporate our cross mutliplication scheme:

(2)   \begin{equation*} \frac{15}{100}=\frac{x}{390}\end{equation*}

Of course, x represents the number we need to know.

Now we do the actual cross-multiplication:

(3)   \begin{equation*} 15(390)=100x\end{equation*}

Dividing out 100 from both sides, we get

(4)   \begin{equation*} \frac{15(390)}{100}=x\end{equation*}

Finally, we arrive at

(5)   \begin{equation*} 58.5=x\end{equation*}

We discover that 15% of 390 is 58.5.

Some other uses of cross multiplication with percents will be covered in future posts. For now, go back to the sun:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

French: practical hints for typing accents

When you tutor French, a student might ask how to produce the accents on an English keyboard.  Here are a couple of options:

On a Microsoft product, every French accent has its own Alt+(4 digit code).  For instance, this ç was typed using Alt+0231.  You need to hold down the Alt key, then type 0231 using the numeric keypad to the right – not the numbers across the top. Here are some codes that, once memorized, can really speed up your French typing:

é:     Alt+0233

è:     Alt+0232

à:     Alt+0224

ç:     Alt+0231

î:     Alt+0238

ô:    Alt+0244

Another way to produce accents is to use the character map.  Look under All Programs→Accessories→System Tools and you should see it.  It’s a grid of different characters which you can copy and paste to your work – really a great tool.

If you’re in Word, of course, you can go Insert→Symbol to find everything you need.  Word has its own shortcut sequences; my wife uses them all the time.  However, the sequences above will work in Word as well.

If you like the copy and paste method, here’s a time saver:  Copy and paste all the accents you’ll need when you first start, like so:

à  ç  è  é  î  ô

Now, you don’t have to flip back and forth between a menu and your work; you can just copy and paste from the list.

I hope this helps:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: What is a “perfect number?”

While enjoying this surprisingly fine summer, the math tutor recalls a definition.

When I was a kid, I read in one of my math texts (it might have been grade six) that a “perfect number” is one whose factors (except itself, of course) sum to it.

Example:  6 is a perfect number.

Factors of 6:    1, 2, 3, 6

Sum of factors (excluding 6 itself):  1+2+3=6

Example:  28 is a perfect number.

Factors of 28:   1, 2, 4, 7, 14, 28

Sum of factors (excluding 28 itself):   1+2+4+7+14=28

I always thought it was an interesting definition.  Curiosities like this are great; they keep you thinking about math when you don’t have to do any:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: What is a Radian?

When you tutor math, you explain radians every semester to your grade 12 students.

Most people begin measuring angles in degrees.  However, you can also measure an angle in radians.  1 rad≈57.3°.

While degrees come from (I am told) Babylonia, or one of the ancient civilizations of that area, radians are a “natural” way to measure angles.  Behold:


In the above picture, CA is a radius. The arc from A to B is the same length as CA. Therefore, angle ACB is 1 radian. 1 radian is the angle that you traverse by following an arc the length of the radius. Said another way, it’s the angle subtended by an arc one radius long.

Recall that the circumference of the circle is 2πr, where r is the radius.  Since 2πr is the exact circumference, 2π radians is exactly 360°.

Radians can be referred to as rads, but are usually stated without any unit. That’s how you can tell which way the angle is measured:  if it’s in degrees, it will have a degree sign.  If it’s in rads, it won’t have any units.  Therefore, an angle of 54° means, of course, 54 degrees.  However, an angle of 32 means 32 rads.

Please keep enjoying this fine summer!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Practical math: Some easy conversions from metric to imperial

As a math tutor, you realize that conversions from metric to imperial are part of the grade 10 curriculum.  Let’s talk about a few that don’t need a calculator.

Even though the (Canadian) high school student grows up in a metric environment, the trades use both systems.  Moreover, the tutor likely grew up in the 70s, so still thinks as much in imperial as metric.

With a calculator, of course, you can easily convert any measurement to any other. Nowadays, you can just key a measurement into your browser and it will return the conversion.  In summer, however, such questions seem to arrive more often in everyday life – possibly when you’re not at your computer.

So, both for those in summer school, as well as those who might find these tricks useful in every day life, here are some simple conversions you can do in your head. While not exact (I think the temp conversion is), they get you within 2% of the answer.

kg to pounds:  double it, then add 10% of the answer.

example:  77 kg to pounds

step 1:  double the mass in kg:  77 times 2 = 154.

step 2:  add 10% more.  15.4 + 154= 169.4

So, 77kg is 169.4lbs.

metres to yards:  just add 10%.

55m is 55 + 5.5 or 60.5 yards.

inches to cm:  multiply by 5, then divide by 2.

4 inches = 5(4)÷2 = 10cm.

Fahrenheit to Celsius:

This conversion comes up a lot, but there is no convenient way without a calculator. You subtract 32 from the Fahrenheit, then divide by 1.8.

Example:  Convert 80F to C

step 1:  80-32=48

step 2:  48÷1.8=27 (rounded to the nearest whole degree).

So, 80F is 27C.

Here’s an irony about summer measurements:  According to Wikipedia, the Canadian football field is 110 yards, whereas the American is 100.  However, the Canadian football field is 100 m (since going from metres to yards you just add 10%).  So the American and Canadian are both 100 long in their own units.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.