Two Common Computation Traps : "Divide by 1/2" & Misjudging the Value of Negative Numbers

These are two very common computation traps that can throw off even the most capable math students.

1) Dividing by ½

It's easy to think that some quantity divided by a half means that the result will be the same as dividing by 2. But, that's not the case at all.

Example: 8 ÷ ½ = ?

Many might confuse the above example with 8 ÷ 2 and think the answer is 4. But. dividing by ½ actually means to multiply by 2, which is the reciprocal of ½.

The confusion may stem from the strong connotation the word "half" has in our everyday usage. For instance, when we say "I'd like half of a bagel," or "I ate only half of the pie," we are, in fact, conveying that a whole was separated at some point into two equal parts and we consumed only one of the two equal parts. As such, we automatically associate the word "half" with division.

But the phrase "divide by half" doesn't mean "divide by 2." These often-conflated phrases mean two very different things. And "divide by half" actually means to multiply by 2.

Hence, 8 ÷ ½ ≠ 4.

8 ÷ 2 = 4.

And, 8 ÷ ½ = 16

2) Misjudging Value of Negative Numbers

When dealing with negative numbers, the number that may look the biggest may often have the least value.

Example: (2021 Test B) Question 78.

Given: -1/49, -49, -1/7, -1

Which number shown above is the greatest?

E) -1/49

F) -49

G) -1/7

H) -1

Of the given numbers, the greatest number isn't -49. In fact, -49 is the smallest number of the four given numbers. To make negative numbers less confusing, it sometimes helps to draw a number line with negative numbers to the left of zero and positive numbers to the right of zero.

<--|-------|-------|-------|-------|-->

-3 -2 -1 0 1

On such a number line, it becomes easier to see that those negative numbers that are closer to 0 actually have a higher value than those that are farther from 0. As such, of the four given numbers, the number that is closest to 0 on a number line is -1/49. That number is only 1/49 less than 0 and is greater than the other three numbers.

Put in math terms:

-49 < -1 < -1/7 < -1/49 < 0

By paying closer attention to the concept of dividing by a half or another fraction and the concept of the comparative value of negative numbers, we can avoid common errors and assure that we maximize our SHSAT math score.