Write the two missing values to make these equivalent fractions correct. Something thirds equals eight twelfths equals four something-ths.
In this problem, we’re given three equivalent fractions. We know this because it says so in the question, but also because of the two equal signs in between them. These equal signs show us that even though fraction may have a different top number or numerator and a different bottom number or denominator, they can still be worth the same thing. And each of these three fractions, we know, are worth the same.
Two good problem solving questions that are really useful are: what do I know and how can I use it to help? And we can use these two questions to help us answer this particular problem. We don’t know the two missing values. But what do we know? We know that both fractions are equivalent to or worth the same as eight twelfths. How can we use this to help?
There are two things to remember when finding equivalent fractions. The first is that we can multiply or we can divide the numerator and the denominator. We can’t add or subtract. The second thing we need to remember when we try to find equivalent fractions is that we need to treat both numbers in the fraction the same way. Whatever we do to the top number or the numerator, we also have to do to the bottom number or the denominator. So let’s use these rules to help us to find the two missing values.
We’ll start by thinking about the first fraction. The denominator we already know is three. The first fraction is an amount of thirds. What is three being multiplied by so the fraction becomes twelfths? We could also ask ourselves the opposite question. How do we get from twelfths to thirds? What do we divide 12 by to get an answer of three? 12 divided by four equals three. Well, thinking about it in the other direction, three multiplied by four equals 12. Either way we know that we’re multiplying or dividing by four.
Remember our second rule, we must always treat both numbers the same. So our missing value must be multiplied by four to give an answer of eight. What number when we multiply it by four gives us eight? How many fours are in eight? Two fours are in eight. So our first fraction is two-thirds. Two-thirds equals eight twelfths.
Let’s think about our final fraction. We don’t know what the denominator is. But what do we know? We know that to convert the equivalent fraction, we’ve changed the numerator from eight to four. How do we get from eight to four using either multiplication or division? Eight divided by two equals four. Or if we think about it in the opposite direction, four multiplied by two equals eight. Either way, we’re dealing with the number two, multiplying and dividing by two. Let’s keep going from left to right and we’ll divide by two. How can we use this information to help us to find the missing value?
What we know, to keep the fractions equivalent, we must treat the numerator and the denominator the same way. The numerator has been divided by two. So the denominator must also be divided by two. 12 divided by two equals six. And so eight twelfths is also the same as four-sixths.
We found our missing values by remembering that we must always treat the numerator and the denominator the same way and by thinking about what we need to multiply or divide by to get from eight twelfths to the numbers we’ve already been given. Two-thirds equals eight twelfths equals four-sixths. The missing values were two and six.