A ratio is the comparison of two quantities of the same kind, or the relationship of one similar quantity to another. Ratios can be written in three different ways using ratio symbols or words, while keeping the same meaning. That is, ratios can be written by placing a fraction bar, the word "to", or the ratio symbol '' : '' between the two values being compared.
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Using the previous example of comparing the two groups of students in Ms. Jones' class, the ratio of ninth grade students to tenth grade students can take on any of the three following forms:
- {eq}\dfrac {10} {20} {/eq}
- 10 to 20
- 10 : 20
All three ways, written using different ratio symbols, compare the 10 ninth graders to the 20 tenth graders in the geometry class, and they all have the same meaning.
Depending on the information needed, ratios can be used to compare two portions within a whole set of data which is called a part-to-part ratio, or a ratio can compare one part of a set of data to the total collection of data which is called a part-to-whole ratio. When we refer back to the geometry class example, comparing the ninth graders to the tenth graders is a part-to-part ratio. The two grade levels are parts of the entire student roster. When the number of ninth graders is compared to the entire student roster, a part-to-whole ratio is created. The ninth graders are part of the whole class.
Ratios show the relationship between two values, but they also have a mathematical connection with division. Remember that the fraction bar is one of the ratio symbols used to write ratios. The fraction bar also means division. So when we look back at the 10 ninth graders and 20 tenth graders, division can be implied with this ratio. Basically, the ninth graders would be divided, shared, or grouped equally with the tenth graders.
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Ratios can be expressed in several different, but equivalent, forms such as fractions, decimals, and percents. The ratio of ninth graders to the total number of students in the geometry class, 10 to 30, is a perfect example to demonstrate how we can express ratios in different and equivalent ways. Here the equivalent expressions are as follows:
- {eq}10:30 {/eq} (Use of the ratio symbol)
- {eq}\dfrac {10}{30} = \dfrac {1}{3} {/eq} (Use of the fraction bar and simplification of the fraction)
- {eq}\dfrac {1}{3} = 0.333... {/eq} ( Simplifying the ratio, and performing the indicated division, 1 divided by 3, gives the ratio in decimal form, which results in a repeating decimal )
- {eq}0.333... = 33.3... \% {/eq} ( Rewriting the repeating decimal as a percent by multiplying it by 100 gives the ratio in percent form )
This demonstrates that ratios have multiple ways of being expressed and they are all equivalent.
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Let's take a look at a few situations in which ratios can be used to express different scenarios or solve different problems.
Example 1: A survey stated that 1 out of 3 students complete their homework on time. Write this as a ratio, and explain its meaning.
Solution: 1:3, 1 to 3, or {eq}\dfrac{1}{3} {/eq} are three different ways to write this ratio, and they all demonstrate that the number of students that complete their homework on time is 1 compared to a total of 3 students surveyed.
Example 2: A certain daycare facility has 15 infants and 20 toddlers. What is the ratio of infants to toddlers?
Solution: 15:20, 15 to 20, or {eq}\dfrac{15}{20} {/eq} can all be used to represent this ratio. The ratio compares the number of infants to the number of toddlers in the daycare facility. Due to the fact that the numbers involved in this ratio share a greatest common factor of 5, the ratio can be simplified by dividing both the numbers in this ratio by this greatest common factor of 5.
{eq}\begin{align} \dfrac{15}{20} &= \dfrac{15\div 5}{20\div 5}\\[0.3cm] &= \dfrac{3}{4} \end{align} {/eq}
We see that we can represent the ratio of infants to toddlers as 15:20 or as 3:4. These are called equivalent ratios, because they have the same overall meaning and value.
Example 3: A runner has a steady workout of 2 miles every 4 days. How many miles does the runner cover in 2 days?
Solution: This example is a good demonstration of how equivalent ratios and equivalent fractions go hand in hand. If we write the ratio in fraction form, then we will be able to see how equivalent fractions will help us answer this question.
{eq}\dfrac{2 \text{ miles}}{4 \text{ days}} = \dfrac{? \text{ miles}}{2 \text{ days}} {/eq}
Working with the ratio in fraction form, we can see that if we divide the 4 days by 2, then we get 2 days. In order to keep the fraction balanced and the above equation true, we also divide the number of miles by 2. This gives the following:
{eq}\begin{align} \dfrac{2 \text{ miles}}{4 \text{ days}} &= \dfrac{2\div 2 \text{ miles}}{2\div 2 \text{ days}}\\[0.3cm] &=\dfrac{1 \text{ miles}}{2 \text{ days}} \end{align} {/eq}
The number of miles the runner covers in 2 days is 1 mile.
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Scaling ratios are mathematical calculations that involve increasing or decreasing quantities at the same time by a specific constant factor. For instance, the ingredients in a recipe are sometimes written as ratios, and there are times when a recipe needs to be increased or decreased by a certain factor in order to get the recipe right for a specific amount of that recipe. Here are few examples.
- The original recipe for 1 loaf of banana bread calls for 3 cups of flour for every 2 cups of sugar. We want to make 2 loaves of bread, how much flour and sugar do we need?
We can determine how much flour and sugar is needed for two loaves of bread by first expressing the ratio of flour to sugar in the original recipe for one loaf, and then doubling both values in the ratio:
3 : 2 (original ingredients)
6 : 4 (double or twice the ingredients)
Thus, to make 2 loaves of bread, we would need 6 cups of flour and 4 cups of sugar.
- What if it is the same recipe, but we only want {eq}\dfrac{1}{2} {/eq} a loaf of bread?
In the same way that we doubled the amount of each ingredient in our original ratio for 2 loaves of bread, we can halve the amount of each ingredient in our original ratio to determine how much is needed for half a loaf of bread.
3: 2 (original ingredients)
{eq}1\frac {1}{2} {/eq} : 1 {eq}\left(\dfrac{1}{2} \text{ the ingredients}\right) {/eq}
Thus, to make half a loaf of bread, we would need {eq}1\dfrac{1}{2} {/eq} cups of flour and 1 cup of sugar.
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A ratio is the comparison of two quantities of the same kind, or the relationship of one similar quantity to another. A ratio can compare two portions within a whole data set, which is called a part-to-part ratio, or a ratio can compare one part of a data set to the total collection of data, which is called a part-to-whole ratio.
Ratios can be written with 3 different symbols:
- '' : '' between the numbers (For example, 10:30)
- ''to'' between the numbers (For example, 10 to 30)
- A fraction bar between the numbers {eq}\left( \text{For example, }\dfrac{10}{30}\right) {/eq}
Ratios can also be expressed in the following equivalent forms:
- A fraction {eq}\left(\text{For example, }\dfrac{10}{30}\right) {/eq}
- A decimal (For example, 0.33... )
- A percent (For example, 33.3%
Since ratios can be written as fractions, there are many similarities between ratios and fractions. Ratios can be simplified to equivalent ratios just like fractions can be simplified to equivalent fractions, by dividing the numerator and denominator by the greatest common factor. Equivalent ratios can also be calculated by multiplying both parts of the ratio by the same number.
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Video Transcript
Definition
A ratio compares two values. It shows you that when you have this much of something, you will need to have that much of something else.
You see ratios used in cooking and when working with model toys. The recipe for hummingbird food, for instance, calls for 4 parts of water for every part of sugar. What this ratio tells you is that however much sugar you put in, you need to put in 4 times as much water. If you use 1 cup of sugar, you need 4 cups of water. If you use 1 tablespoon of sugar, you need 4 tablespoons of water.
Writing Ratios
When you want to write a ratio mathematically, there are three specific formats you can choose from. You can choose to write it in colon form with a colon separating the two numbers. Or you can choose to use the word 'to' in between the numbers. Also, you can write it as a fraction. All are valid and all mean the same thing. When you write it as a fraction though, you will want to keep it in fraction form even if you can simplify it further. As in the hummingbird food recipe, the ratio in fraction form would stay 4/1 even though it can be simplified to just 4.
Using Ratios
Use ratios to describe any two things that have a fixed value when compared to each other. The dimensions of photos, for example, have a fixed value when you compare the length to the width or vice versa. A photo that has a width of 5 and a length of 7 will have a width to length ratio of 5:7. A larger photo, such as one with a width of 8 and a length of 10, has a ratio of 8:10.
Another real world application of ratios is in the building of model toys. Many model toy cars come scaled at 1/12th of life size. What this means is that every measure of the toy car equals 12 measures in real life. What measures 1 cm in the toy car will measure 12 cm in the real world.
Scaling Ratios
In many ratio problems, you will need to scale your ratios to find your answer. Going back to the model toy car, if you had the measurements of the real car in front of you, you can use the ratio to calculate the measurements of the toy car by multiplying the real world measurements by the ratio. If the real tire measured 61cm in diameter, then the toy tire will measure 5.08 cm. This is because 61 x (1/12) = 5.08. Whatever real world dimensions you had will just need to be multiplied by the ratio to get the toy dimensions.
You can also scale ratios to find new measurements that maintain the same ratio. Let's say you had a photo whose dimensions are 5 inches wide by 7 inches long. The ratio of this photo is 5 to 7. We don't write dimensions in a ratio form because dimensions are not ratios but exact measurements and are written with either a multiplication symbol or the word 'by.' You like how the picture's width is that much shorter than its length with a ratio of 5:7. You can use this ratio to figure out other possible dimensions that will maintain the same width to length ratio.
To do this, you first figure out how much bigger or smaller you want one of your dimensions to be. Let's say I want the width to be 10 inches wide instead of 5. Well, 10 inches is 2 times bigger than 5. I'd have to multiply the 5 by 2 to get to 10. So, to figure out my length, I'd have to multiply it by the same amount, by 2. My new dimensions then are 10 inches by 14 inches to maintain the same ratio of 5 to 7.
Lesson Summary
Ratios are commonly used when cooking and when describing dimensions. There are three ways to write ratios. One is with a colon, the second is with the word 'to' and the third is as a fraction. Scaling a ratio requires the multiplication of the same number to both values.
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