We've covered how to see which of two integers is greater (one way is by looking on a number line and seeing which number is further to the right). But what happens if it's harder to find the numbers in question on that number line? Like when we're working with fractions and decimals...
Question
Which of each pair is larger:
Answer
- 0.5, 0.55
Analysis
- 0.5 < 0.55
Question 1
When looking at which fraction is bigger, such as the fractions
, we need to compare fractions that have common denominators. Let's talk about why.We talked already about how the bottom number (the denominator) is the number of steps we make from 0 to 1 on a number line and the top number (the numerator) is the number of steps we've made between 0 and 1. With
, we're comparing how far we've gone needing two steps with how far we've gone needing 3 steps - unless you just happen to know how far each is and where they sit on the number line, it's hard to figure out like that! So let's set up the fractions so that they are in a form where they both have an equal number of steps - they both have the same denominator.Let's first look at the number line and see where the two fractions end up:
The red line shows us getting to 1/2 and the red dot is where we land. The blue lines show the two hops we've taken out of the three we'd need to get to 1. Clearly, the second blue dot it further right than the red dot, and so
, or 1/2 is smaller than 2/3 (or said another way, 2/3 is bigger than 1/2).But what if we could do this problem and have both fractions have an equal number of jumps to get to 1 - or in other words, have the same denominator. We can do that.
We'll cover how we do it in the entry Multiplication, Division, and Rational Numbers, but for right now I'll say that we can have both fractions make the trip from 0 to 1 in 6 jumps. Let's look at how that would work.
I'll keep the graph above and I'll put the 6 little jumps from 0 to 1 underneath in purple:
If you count, you'll see that the 1/2 in red is the same as 3/6 in purple - 1 jump out of 2 in red is the same as 3 out of 6 jumps in purple. Also, 1 jump out of 3 jumps in blue is the same as 2 out of 6 jumps in purple and 2 jumps out of 3 in blue is the same as 4 out of 6 in purple. And so we can say that:
And so:
Question 2
When we're looking at decimals, we can approach the "which is bigger" question in a couple of different ways:
One way is to think about how far we've gone from 0 to 1 while strictly in decimal form. Let's first remember that the first digit to the right of the decimal point is the tenths place (image from www.themathpage.com):
If we go 0.5, we've gone 5 tenths.
If we go 0.55, we've gone 5 tenths plus we've gone another 5 hundredths - so we've gone farther along the path from 0 to 1:
The red path gets us to 0.5 (which is the same as 1/2 - we'll talk about that in a later entry) while the green path is the additional 5 hundredths. Therefore,
0.5 < 0.55
Another way to do this problem is to convert the question to fractions.
Remember that for 0.5, we say that that number is 5 tenths. This means that we've moved 5 jumps along the 10 needed to go from 0 to 1. And that is the same as this fraction:
For 0.55, we can say it is 5 tenths and 5 hundredths, or we can say it's 55 hundredths - it means the same thing. We've moved 55 jumps out of the 100 needed to move from 0 to 1. And that is the same as this fraction:
Now we need to do one more step to make the fractions comparable, and that is to convert
to a form that has 100 as its denominator. It turns out that 
. And so:
and therefore:
0.5 < 0.55
Vocabulary used:
For more information check out these links (comment to add your favourite link):
https://www.themathpage.com/arith/compare-fractions-2.htm
Where might you have come from?
Fact-orials Index
Numbers:
Associated Operations:
Relations:
Graphing:
Where might we go?
Operations with different kinds of numbers:
A useful resource for comparing fractions:
ReplyDeletehttps://www.themathpage.com/arith/compare-fractions-2.htm