Ratio And Proportion

What Is a Ratio?

A ratio is a mathematical way to compare the size of two or more quantities. It tells us how much of one thing there is compared to another thing. Ratios are everywhere: in recipes, on maps, and in sports statistics. They help us understand the relationship between different numbers.

For example, imagine you have a bowl of fruit with $5$ apples and $3$ oranges. We can express the relationship between apples and oranges using a ratio. There are three common ways to write a ratio:

• Using a colon: $5:3$
• As a fraction: $\frac{5}{3}$
• Using words: "5 to 3"

All three forms mean the same thing: for every $5$ apples, there are $3$ oranges. The order of the numbers in a ratio is very important. The ratio of apples to oranges is $5:3$, but the ratio of oranges to apples is $3:5$. They are not the same!

We can also use ratios to compare a part to the whole. In our fruit bowl, there are $5$ apples and $3$ oranges, so there are $5+3=8$ pieces of fruit in total. The ratio of apples (a part) to the total fruit (the whole) is $5:8$ or $\frac{5}{8}$.

How Do You Simplify Ratios?

Just like fractions, ratios can often be simplified to make them easier to understand. Simplifying a ratio means dividing both parts of the ratio by their greatest common factor (GCF). The GCF is the largest number that divides into both numbers without leaving a remainder. A simplified ratio is equivalent to the original ratio.

Let's say a school has $18$ teachers and $270$ students. The ratio of teachers to students is $18:270$. This is a correct ratio, but the numbers are large. Let's simplify it by following these steps:

1. Write the ratio as a fraction: $\frac{18}{270}$
2. Find the Greatest Common Factor (GCF) of the two numbers. The factors of $18$ are $1,2,3,6,9,18$. The factors of $270$ are many, but we can see that both numbers are divisible by $9$ and also by $2$, so they must be divisible by $18$. The GCF is $18$.
3. Divide both parts of the ratio by the GCF.
   $18÷18=1$
   $270÷18=15$
4. Write the new, simplified ratio. The simplified ratio is $1:15$ or $\frac{1}{15}$.

This simplified ratio tells us that for every $1$ teacher, there are $15$ students. This is much easier to grasp than $18:270$, even though they represent the exact same relationship.

What Is a Proportion?

A proportion is a statement that two ratios are equal. It's essentially an equation involving two equivalent fractions. If you can simplify two ratios and they become the same, then they form a proportion.

For instance, we know that $\frac{1}{2}$ is equal to $\frac{2}{4}$. So, we can write the proportion: \frac{1}{2} = $\frac{2}{4}$. We read this as "one is to two as two is to four."

How can we check if two ratios form a proportion? There are two main methods:

1. Simplifying: Simplify both ratios to their simplest form. If they are identical, they form a proportion. For example, to check if $6:9$ and $10:15$ are proportional, we simplify both. $6:9$ simplifies to $2:3$ (dividing by GCF of $3$). $10:15$ simplifies to $2:3$ (dividing by GCF of $5$). Since both simplify to $2:3$, the ratios are in proportion.
2. Cross-Multiplication: This is a very powerful method. For two ratios $\frac{a}{b}$ and $\frac{c}{d}$, if their cross-products are equal, they form a proportion. The cross-products are $a\times d$ and $b\times c$.

Let's use cross-multiplication on our last example, $\frac{6}{9}$ and $\frac{10}{15}$.
$6\times 15=90$
$9\times 10=90$
Since the cross-products are both $90$, the ratios form a proportion. Proportions are the key to solving for an unknown quantity in a ratio problem.

How Do You Solve Proportions?

Often, you will be given a proportion with one number missing, and you'll need to find its value. We call this missing number a variable, often represented by a letter like $x$. The best way to solve for this unknown value is by using cross-multiplication.

The rule for cross-multiplication is simple and reliable. It turns the proportion into a simple algebra equation that is easy to solve.

Here are the steps to solve a proportion:

1. Set up the proportion: Write the two equal ratios, with the unknown value represented by a variable (e.g., $x$).
2. Cross-multiply: Multiply the numerator of the first ratio by the denominator of the second ratio. Then, multiply the denominator of the first ratio by the numerator of the second. Set these two products equal to each other.
3. Solve for the variable: Use division to isolate the variable and find its value.

This method is incredibly useful for solving a wide variety of problems, from adjusting recipes to calculating distances from a map.

How to Use Proportions in Word Problems

The real power of proportions comes alive when solving word problems. The key is to correctly identify the relationship being described and set up the proportion carefully. The most important rule is to be consistent with the units in your ratios.

For example, if you set up your first ratio as $\frac{\text{miles}}{\text{hours}}$, your second ratio must also be $\frac{\text{miles}}{\text{hours}}$. You cannot mix it up and use $\frac{\text{hours}}{\text{miles}}$ for the second one.

Let's break down the process for tackling a word problem:

1. Read and Understand: Identify the two quantities being compared in the problem.
2. Set Up the First Ratio: Use the information given in the problem to write the first ratio. Label your units!
3. Set Up the Second Ratio: Write the second ratio, using a variable (like $x$) for the value you need to find. Make sure the units are in the same position (numerator/denominator) as your first ratio.
4. Write the Proportion: Set the two ratios equal to each other.
5. Solve and Check: Cross-multiply and solve for the variable. Does your answer make sense in the context of the problem?

What Are Unit Rates?

A unit rate is a special type of ratio where the second quantity is $1$. It simplifies a ratio to show how much of the first quantity corresponds to just one unit of the second quantity. This makes comparisons much easier.

You see unit rates all the time:

• Miles per hour: The number of miles traveled in $1$ hour.
• Price per pound: The cost for $1$ pound of apples.
• Words per minute: The number of words you can type in $1$ minute.

To find a unit rate, you simply divide the first quantity by the second quantity. For example, if you travel $150$ miles in $3$ hours, your unit rate (speed) is:
$$\frac{150\text{ miles}}{3\text{ hours}}=50\text{ miles per hour}$$

Unit rates are extremely helpful for being a smart shopper. By calculating the price per unit (like price per ounce or price per item), you can determine which product offers the best value, or the "best buy."

Finding the Best Buy

Imagine you are at the store and want to buy juice. You see two options:

To find the better deal, we calculate the unit price (cost per ounce) for each brand.

Brand A:
$$\frac{\mathrm{\$}2.88}{32\text{ oz}}=\mathrm{\$}0.09\text{ per ounce}$$

Brand B:
$$\frac{\mathrm{\$}3.36}{48\text{ oz}}=\mathrm{\$}0.07\text{ per ounce}$$

Comparing the unit rates, Brand B costs only $7$ cents per ounce while Brand A costs $9$ cents per ounce. Therefore, Brand B is the better buy.

Common Mistakes to Avoid

Ratios and proportions are straightforward once you get the hang of them, but a few common mistakes can trip you up. Being aware of them is the first step to avoiding them!

• Mixing up the order. The ratio $a:b$ is not the same as $b:a$ (unless $a=b$). When you set up a proportion, make sure the quantities in the numerators match (e.g., both are 'cups of sugar') and the quantities in the denominators match (e.g., both are 'number of cookies').
  Incorrect: $\frac{\text{sugar}}{\text{cookies}}=\frac{\text{cookies}}{\text{sugar}}$
  Correct: $\frac{\text{sugar}}{\text{cookies}}=\frac{\text{sugar}}{\text{cookies}}$
• Incorrect cross-multiplication. A common error is to multiply the numerators together and the denominators together. Remember to multiply diagonally across the equals sign. For $\frac{a}{b}=\frac{c}{d}$, it's $a\times d$ and $b\times c$, not $a\times c$ and $b\times d$.
• Forgetting to simplify. While an unsimplified ratio like $25:100$ is technically correct, the simplified version $1:4$ is much easier to work with and understand. Always simplify your final answer when possible.
• Confusing part-to-part with part-to-whole. Read the question carefully. If there are $5$ dogs and $6$ cats, the ratio of dogs to cats (part-to-part) is $5:6$. The ratio of dogs to total pets (part-to-whole) is $5:11$. These are very different comparisons.

Quick Summary

Here's a quick review of the most important concepts about ratios and proportions:

• Ratio: A comparison of two quantities, written as $a:b$, $\frac{a}{b}$, or "a to b".
• Simplifying Ratios: Divide both parts of the ratio by their Greatest Common Factor (GCF).
• Proportion: An equation stating that two ratios are equal, such as $\frac{a}{b}=\frac{c}{d}$.
• Solving Proportions: The best method is cross-multiplication. If $\frac{a}{b}=\frac{c}{d}$, then $a\times d=b\times c$.
• Unit Rate: A ratio where the second quantity is $1$. To find it, divide the first quantity by the second. It's great for comparing prices.
• Consistency is Key: When setting up proportions, always keep the units in the same order in both ratios.