# Implied Ratios: Definition & Examples

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Implied ratios show up often in our daily lives. Let's take a look at what implied ratios are and how to solve implied ratio problems. We will use real world examples of these implied ratios to solidify our understanding of the concept.

## Implied Ratio

Suppose you are at the store buying cashews. The sign at the store says that 2 pounds of cashews cost \$15.

Notice, the sign advertising the cost of cashews displays the following:

\$15 / 2 lbs.

Believe it or not, this is actually a mathematical display! In mathematics, we would call this expression a ratio. A ratio is a comparison of two quantities that displays the size of the quantities relative to one another. In this example, we see that the ratio \$15 / 2 lbs. compares the price of the cashews relative to 2 lbs.

Of course, the sign doesn't explicitly say that this is a ratio, so we call this an implied ratio. An implied ratio is a ratio that is implied even though it's not presented explicitly as a ratio. You may be wondering what is the point of these implied ratios! Actually, problems involving implied ratios show up quite often in the world around us!

For instance, we know that 2 pounds of cashews costs \$15, but suppose you only want to buy 1 pound of cashews. This is an implied ratio problem because it involves ratios, even though it doesn't come outright and say that it involves ratios.

Because implied ratio problems don't explicitly state that they are dealing with ratios, it is good to be able to recognize these types of problems so we can solve them appropriately. Speaking of which, let's take a look at how to solve these types of problems!

## Solving Implied Ratio Problems

Solving implied ratio problems isn't too difficult. It simply involves setting up a proportion and solving for the unknown. Great, but what is a proportion? A proportion is two ratios set equal to each other.

We see that a proportion takes on the form

a / b = c / d

where a / b and c / d are ratios. Well, that's easy enough! Furthermore, we can simplify and solve for unknowns within a proportion using a method called cross multiplication.

Cross multiplication gives that if a / b = c / d, then ad = bc. All this information is really going to come in handy when solving implied ratio problems! Let's take a look! To solve implied ratio problems, we use the following steps;

1. Identify the given implied ratio in the problem.
2. Identify the unknown quantity in the problem with a variable and set up a ratio involving that unknown quantity with the information given in the problem.
3. Set the two ratios found in steps 1 and 2 equal to each other, forming a proportion.
4. Solve the proportion for the unknown quantity using cross multiplication.

Alright! Let's put these guys into action!

## Examples

Back to our shopping trip! We know that 2 pounds of cashews cost \$15, and we want to know how much 1 pound of cashews cost. The first step in solving this problem is to identify the given ratio in the problem. We've actually already done that with the ratio \$15 / 2 lbs.

The second step is to identify the unknown quantity with a variable, and set up a ratio. In this problem, the unknown quantity is the cost of 1 pound of cashews, so let's represent that with the variable c. To set up a ratio including this variable, we again compare cost to pounds. Thus, we have \$c / 1 lb. It is important to note that we always want the units in the numerators to match in the two ratios and the units in the denominators to match in the two ratios.

Okay, on to the third step! We now go ahead and set up a proportion by setting our two ratios equal.

That's probably the easiest step! Lastly, we simply solve the proportion for the unknown using cross multiplication.

We get that c = \$7.50, so we know that 1 pound of cashews cost \$7.50! Pretty neat, isn't it?

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