# How can I explain ratio problems without talking about ratio?

My students haven't learned about ratio yet, but in the test, there are questions that relate to ratio. I'm not sure how to explain this kind of problem without talking about ratio.

For example,

1. every 100g soup contains 150kj energy, one serve of the soup contains 450kj energy, what is the mass of one serve of the soup?

2. A 8mm * 8mm square computer chip is represented by a 4cm * 4cm graph, what scale is used in the drawing?

How would you explain this kind of problem without talking about ratio? Thank you!

• You posted this on the meta site associated to the Mathematics Educators site. It seems to me this as a question for the main site, which is why I moved it there. – quid May 7 '17 at 11:49
• Thank you @quid no wonder I couldn't find the right tag. – EmmaXL May 7 '17 at 11:57
• My students haven't learned about ratio yet, but in the test, there are questions that relate to ratio. What is "the test," and why is your curriculum not aligned with it? How would you explain this kind of problem without talking about ratio? Are you subject to some kind of micromanagement that prevents you from talking to your students about a certain topic? This seems like a problem that was created by administrative constraints and has an administrative solution, not an educational one. – Ben Crowell May 8 '17 at 1:14
• @BenCrowell The teacher might have perfectly valid educational reasons to avoid introducing a new concept. Ratios are quite confusing, even for me: the ratio $2:3$ (like $2$ cups of sugar to every $3$ cups of flour) has a very subtle relationship to the fraction $2/3$ (namely that you have $2/3$ of a cup of sugar for every $1$ cup of flour). In fact, ratios are not fractions: it doesn't really make sense to add them. From a mathematically sophisticated point of view a ratio is an element of a projective space. For instance, the ratio 2:3:5 is equivalent to the ratio 4:6:8. Continued... – Steven Gubkin May 8 '17 at 12:33
• @BenCrowell So "really" the ratio 2:3:5 is the point [2,3,5] in $\mathbb{RP}^3$. Note that the ratio $2:0$ is perfectly valid, while $2/0$ is not a rational number. For these reasons, the teacher might want to really drive home the meaning of rational numbers and division before introducing the concept of ratio. – Steven Gubkin May 8 '17 at 12:36

Have your students covered division and multiplication yet? To me, these problems can all be solved using the "meaning of division" and the "meaning of multiplication".

"Meaning of multiplication": If I have n groups of equal size m, then I have n*m total objects.

"Meaning of division": There are actually two!

(a) If I have m objects which are partitioned into n equally sized groups, then I have m/n objects in each group (this is the "how much in each group" interpretation of division).

(b) If I have m objects, and I put these objects into equally sized groups where each group has n objects in it, then I have m/n groups of objects (this is the "how many groups" interpretation of division).

Here are the answers to your questions from this perspective:

1. Every 100g of soup contains 150kj of energy. Call 100g of soup a "portion". I have a "serving" of soup which contains 450 kj of energy. So, if I want to see how many 100g portions of soup are in one serving, I need to see how many groups of 150kj there are in 450kj. The answer to this question (by "the meaning of division") is 450/150 = 3 portions. Then, since each of these 3 equally sized portions of soup is 100g, (by "the meaning of multiplication") I have 3*100g = 300g of soup.

2. 4cm represents 8mm. So I have 4 centimeters, which, in total, represent 8 millimeters. How many millimeters does one centimeter represent? Each centimeter represents an equal sized group of millimeters. So I have 4 equally sized groups of millimeters. Thus each group has 8/4 = 2 millimeters in it. So each centimeter represents 2 millimeters.