December 2014 / January 2015 Newsflash: Tips & Techniques

Upper Elementary and Middle School

Find the perimeter of the shape below. (Hint: there is enough information.)

Solution:

The length on the bottom of the rectangle is the same as the length on the top, although the top has been split into two pieces. Similarly the width on the left is 11 inches and the width on right is also 11 inches, again in two pieces. 

Now, the length of the side is given in inches and the length of the bottom is given in feet. Since we have to add to get the perimeter, the Law of SAMEness requires the units to be the same. Let’s change 2 feet into 24 inches (2 x 12). 

Note that it is not necessary to know the exact measurement of the length and width. All that matters is that we know the total length of each. 

With that, 11 in + 11 in + 24 in + 24 in = 70 inches.

Algebra 

If ax + 2a = c and x + 2 = 3, express a in terms of c. 

Solution: 

If ax + 2a = c, then a(x + 2) = c. Now, substitute “3” for “x + 2” and you get 3a = c, 

so a = c/3.

October/November 2014 Newsflash: Tips & Techniques

From time to time, we get lucky and the computation we have to do can be solved using a specific technique or trick that allows us to do effortless mental math.

Level 1:

12 x 125 = _______

The trick to this question is to see that “12” can be broken down into “3 x 4.”

            Think of “12 x 125” as “3 x (4 x 125).”

Now 4 x 125 = 500. (4 x 100 + 4 x 25)

Then, 3 x 500 = 1,500.

Try this (mentally): 8 x 175 = ________.

Level 2:

1/5 + 1/6 = __________

When you add two fractions, and the numerators of the fractions are both 1, the numerator of the answer is the sum of the denominators, and the denominator of the answer is the product of the denominators. In this problem, the numerator is   (5 + 6), and the denominator is (5 x 6).

So, 1/5 + 1/6 = 11/30. [Note: If the denominators are not relatively prime, the answer will have to be reduced.]

Try this (mentally): 1/12 + 1/13 = _____________.

Level 3:

2,000 ÷ 16 = ____________

The trick here is to recognize that 16 is a power of 2, and that powers of 2 make division into “half-of-a-half-of-a…”

In this case, since 16 is “2 to the fourth power,” we simply cut 2,000 in half 4 times.

So,

            Half of 2,000 is 1,000,

                        Half of 1,000 is 500,

                                    Half of 500 is 250, and finally,

                                                Half of 250 is 125.

So, 2,000 ÷ 16 = 125.

Try this (mentally): 3,000 ÷ 8 = ____________.

Note that there is more than one trick for each of these questions. See if you can find others!

August/September 2014 Newsflash: Tips & Techniques

“Thinking backwards”—starting at the end of the problem and using reverse operations—can help you solve certain problems at different grade levels.

Upper Elementary and Middle School:

A certain number is doubled. That answer is tripled. Finally, that answer is quadrupled and the answer is 60. What is the original number?

Answer:

When we quadruple a number, we multiply it by 4. As division is the inverse of multiplication, 60 ÷ 4 = 15. Tripling a number means “multiply by 3,” so 15 ÷ 3 = 5. Finally, doubling a number means “multiply by 2,” thus, 5 ÷ 2 = 2.5, or 2 ½.

Algebra:

A certain number is quadrupled. 3 is added to the answer. That answer is then tripled. Finally, when that answer is cut in half, the answer is 12.

a)     Write an equation that describes this problem.

b)     What is the original number?

Answer:

a)   We’re looking for a certain (unidentified) number, x. “x, quadrupled” means “4x.” Then, we add 3 to the answer, so 4x + 3. We then triple the quantity “4x + 3”: 3(4x + 3). Finally, we split the quantity 3(4x + 3) in half in order to yield the final answer, 12, so  {3(4x + 3)} /2 = 12.          

b)     To find the original number, we solve for x, which involves “canceling out” the numbers by using inverse operations.

So, (2) {3 (4x + 3)} /2 = 12 (2) multiply both sides by 2…

{3 (4x + 3)}/3 = 24/3 divide both sides by 3…

4x + 3 – 3 = 8 – 3  subtract 3 from both sides, and…

4x /4= 5 /4  divide both sides by 4.

Thus, x = 1.25

Tips and Techniques: Finding 100% of a Number

Given a percentage of a number, how do you find 100%? We’ll show you different ways to find 100%.  

First, remember that 100% means “all of it… the whole thing.”

Example 1: 25% of what number is 5?

This question asks, “If 25% of a number is 5, then what is 100% of the number (the whole thing)?”

The strategy here is to see how many times the “percent number” (in this case, 25) goes into 100, and then count by that number until we reach 100—the whole thing.

Here, we’re told that 25% of a number is 5. So, to find 100% of the number, we count by 25s up to 100: 25, 50, 75, 100.

25% is 5, so

            50% is 10,

                        75% is 15, and

                                    100% is 20.

Example 2: A man spent $10.00, which was 20% of his money. How much money did he have to start?

If 20% of his money is $10.00, then 40% is $20.00. (20% + 20% = 40%)

Similarly, 60% is $30.00 (40% + 20% = 60%), 80% is $40.00, so 100% is $50.00—the amount of money the man had to start.

Here’s an even faster way to approach this problem. There are 5 20%s in 100%, so this problem calls for five sets of $10.00, which equals $50.00.

Example 3: 10 kids—5% of the kids enrolled at a particular school—are out sick. How many kids are enrolled at the school?

Here, 5% is 10 kids. Since there are 20 5s in 100, we multiply 20 by 10 and get 200, the total number of students at the school.

Try these:

1. 10% of what number is 8?
2. 20% of what number is 4?
3. 20% of what number is 6?
4. 50% of what number is 28?
5. 5% of what number is 11?
6. 1% of what number is 2?