Question

Your little sister is having a party and you are in charge of the party goody bags. You decide that each bag should have 3 candy bars, 1 bottle of nail polish, and 2 pieces of bubble gum. You bought an 18 -pack of candy bars, 12 bottles of nail polish, and 20 pieces of bubble gum. (a) What is the ratio of candy bars to bottles of nail polish to pieces of bubble gum in the goody bags? (b) What is the limiting "goody"? (c) Assuming you have an unlimited supply of bags, how many goody bags can you make? (d) How many of each type of goody will be left over?

Short answer

The ratio of candy bars to bottles of nail polish to pieces of bubble gum in the goody bags is 3:1:2. Candy bars are the limiting 'goody'. Hence only 6 total goodie bags can be made. After making the bags, leftover goodies will be: no candy bars, 6 bottles of nail polish, and 8 pieces of bubble gum.

Step-by-step solution

Calculate the ratio of the items

In each bag there are 3 candy bars, 1 bottle of nail polish, and 2 pieces of bubble gum. Hence the ratio of candy bars to bottles of nail polish to pieces of bubble gum in the goody bags is \(3:1:2\).

Identify the limiting goody

The limiting factor (or goody) is the item that will run out first and thereby decide the number of total bags one can make. To find it, check how many full sets of goody bags can be completed with each type of goody: candy bars can make 18/3=6 bags, nail polish can make 12/1=12 bags, and gum can make 20/2=10 bags. Therefore, since candy bars can make the least amount of bags (6), they're the limiting goody.

Find the total number of goody bags you can make

As identified in the previous step, the candy bars are what limit the total amount of goody bags that can be made. So, the total amount of goody bags that can be produced is the same as the amount yielded by the candy bars, which is 6.

Determine the number of leftover goodies

To determine the leftover amounts of each goody, calculate how many of each goody is used and subtract this from the initial amount. For the candy bars, all 18 will be used (6 bags * 3 bars/bag = 18 bars). The nail polish will have 12 - 6 = 6 left (since only one bottle goes in each bag). For the bubble gum, 20 - 12 = 8 will be left (two pieces go in each bag.

Key concepts

Mathematical Ratios

When we talk about mathematical ratios, we're referring to the relationship between two or more numbers that indicates their relative sizes. Consider ratios as a way to compare quantities, much like fractions, but without the need to have the same denominator. In our example, the goody bags contain 3 candy bars, 1 bottle of nail polish, and 2 pieces of bubble gum, which creates a ratio expressed as 3:1:2.

This ratio tells us for every 3 candy bars, there is 1 bottle of nail polish, and 2 pieces of bubble gum. Ratios are vital in understanding proportional relationships and are commonly used in everyday life, from following recipes to sizing up photographs. It's important to always maintain the order when writing ratios, since the first number corresponds to the first item mentioned, and so forth, ensuring that anyone reading the ratio understands the comparison precisely.

Limiting Factor

Understanding the Limiting Factor

In our context, the limiting factor refers to the component within a process that restricts the amount of the final product—the goody bags, in this case. It's more commonly recognized in other areas such as production and economics where it defines the constraint that prevents achieving higher output.

In our exercise, we identify the limiting factor by dividing the total number of each goody by the number needed per bag. The candy bars were the limiting factor since they caused the cap on the number of bags that could be made. The concept of limiting factors is important for efficient resource management, ensuring that you can plan and allocate appropriately. Without an understanding of the limiting factor, you might overpurchase certain items while running out of others too quickly.

Arithmetic Calculations

Essential Calculations in Arithmetic

Arithmetic calculations form the foundation for higher mathematical concepts and daily problem-solving. These include addition, subtraction, multiplication, and division. In the exercise, we applied these basic calculations iteratively to solve the problem.

To find the limiting factor, we divided the total amount of each goody by the number required per goody bag. To determine the number of leftover goodies, subtraction was used. These operations are straightforward but underpin much more complex work. Mastery of arithmetic calculations allows students to develop a solid understanding of number manipulation and sets the stage for tackling algebra, calculus, and beyond. It cannot be overstated how crucial these basic skills are in every facet of mathematics.