3.4: Word problems for multiplying fractions

In the following, “A/B” represents the fraction “A over B.” For example, “2/3” refers to the fraction “2 over 3” or “two-thirds.”

Word problems for the multiplication problem, A/B x C/D, have the form “A/B of C/D of a unit is how much of a unit?” For example,

• Arithmetic problem: Solve 3/5 x 1/3 (three-fifths times one-third).
• Word problem: If 1/3 of the students in a class have a pet and 3/5 of the students with a pet have a dog, then what fraction of the students in the class have a dog as a pet?
• 3/5 x 1/3 = 1/5, so 1/5 of the students in the class have a dog as a pet.

Here’s a couple more examples:

• Arithmetic problem: Solve 1/4 x 2/3 (one-quarter times two-thirds).
• Word problem: If a recipe calls for 2/3 of a cup of soy sauce, then how much soy sauce is needed for 1/4 of the recipe?
• 1/4 x 2/3 = 1/6, so 1/6 of a cup of soy sauce is needed for 1/4 of the recipe.

• Arithmetic problem: Solve 1/6 x 3/4 (one-sixth times three-quarters).
• Word problem: If Alastair used 3/4 of a cup of butter to make a cake and he ate 1/6 of the cake, then how much butter did he eat?
• 1/6 x 3/4 = 1/8, so Alastair ate 1/8 of a cup of butter.

Pay close attention to the wording for this type of problem:

• Word problem: If 1/5 of the students in a class have a dog and 1/6 of the students in the class have a cat, then what fraction of the students in the class have a cat and a dog?
• Can we solve this by calculating 1/6 x 1/5?
• No, because this assumes that 1/6 of the students in the class that have a dog also have a cat (and we don’t know that). It could be that no students in the class have a cat and a dog!

• Word problem: If Carina used 6/7 of a bag of chocolate chips to make a batch of brownies and then she ate 2/3 of the batch of brownies, then how many chocolate chips did she eat?
• Can we solve this by calculating 2/3 x 6/7?
• No, because this calculates the fraction of a bag of chocolate chips that Carina ate, not the number of chocolate chips she ate.

• Word problem: If Sebastián ate 1/3 of a whole pizza from 3/4 of a pizza left over in the fridge, then how much of a whole pizza did he eat?
• Can we solve this by calculating 1/3 x 3/4?
• No, because this would calculate the fraction of a whole pizza that Sebastián ate if he ate 1/3 of the leftover pizza. But he ate 1/3 of a whole pizza, which gives us the answer to the question directly with no calculation needed.

The video below works through some examples of word problems for multiplying fractions.