Ratios & Proportions

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Overview

Overview – Percents are Ratios, and Equal Percents are Proportions, and Many Types of Problems can be Solved Using a Proportional Relationship Equation.
Ratios
1. Definition of Ratio – Compare 2 Quantities with the Same Unit (e.g. 2 CUPS of sugar for every (1) CUP of milk)
  1. Written 3 ways; don’t need to write unit (e.g. 1:4, 1 to 4, or 1/4)
  2. Compare / Contrast to Fractions,
    1. Ratios must be reduced like fractions
    2. Create equivalent ratios just like equivalent fractions
    3. But…Fractions represent the part of a whole amount, whereas ratios are different things compared using the same unit
2. Working with Ratios
  1. Convert to the same units before expressing a ratio
  2. Order Matters – When analyzing a Ratio Problem, numbers have to be understood exactly as the problem represents them.
  3. A Fraction in my Ratio? Keep Calm and Remember Your Steps!
3. A Special Kind of Ratio – Rate: Compare 2 Quantities with Different Units
  1. Definition of Unit Rate – Shows how much something is per 1 unit of something else (e.g. gas costs $2.78 per 1 gallon; quantities are dollars and gallons)
  2. Common Unit Rates (e.g. pay rate, speed, unit price)
Proportion – A Relationship of 2 Equal Ratios (e.g. 1/2 = 2/4)
1. Identify Proportionality by Cross – Multiplying Numbers in the Ratios
2. Proportions are Just a Mathematical Way to Find a Missing Value
3. Easily Recognize Proportion Problems When you See Them
4. Solve Different Types of Problems Using Proportions
  1. Find the Missing Quantity in a Problem with Ratios
  2. Find the Unit Rate
  3. Percent Problems Solved Using Proportions – Each contains 3 parts (Part, Whole, Percent)
    1. Find the Percent of a Number
    2. Find the Part When the Whole is Given
    3. Find the Whole when the Part is Given

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Welcome to the course Ratios and Proportions. Learning to use ratios and proportions to compare quantities or values is a simple matter of learning specific vocabulary and learning types of problems that require them, yet this skill will quickly make you a more sophisticated math problem solver.

A ratio is a comparison of 2 quantities that have the same unit. When cooking, you can compare the quantities of milk to sugar as 1 to 2. You can compare the number of cats to dogs living in a city, with the unit being whole numbers. You can compare summer and winter rainfall, both using the unit of inches. Its important to understand ratios in life so that you can understand relationships between things. Just reading the news you’ll hear ratios used all the time.

Ratios and fractions seem the same at first, but they aren’t. Ratios can look like fractions if written with the fraction mark. And they’re similar because you must reduce ratios to a form in which both numbers are divisible only by 1. 3 to 6 is 1 to 2. Reduce them the same way you would fractions.

Ratios do have equivalents, like fractions do (½ and 3/6 and 6/12), and this becomes important when you solve problems using ratios. Unlike fractions, though, ratios are a comparison of 2 different quantities, like a comparison of milk to sugar in a recipe, as opposed to a fraction, which is a part of a 1 whole quantity, like ½ cup of milk.

Learn how ratios can be written so you will recognize them in all forms in a newspaper or on a test. A colon, the word “to”, and a fraction mark are all ways you’ll see ratios written. But regardless of how they are written, you always speak them as “1 TO 4”, saying the number on the left first. Ratios do not have units written with them because the context they are written in will tell you what the unit is.

There are a few specific points to remember when solving problems with ratios. When 2 units are compared in a problem, you have to convert 1 unit so they are the same. For example, you can’t compare months to years, so one unit has to be converted to match the other.

When you read a problem and have to work with a ratio, it is important to write the ratio in the right order. So if a problem asks about a ratio in one order, even if the numbers are listed in a different order in the problem, you have to write the ratio according to the question. So here, the answer is 6 to 9, not 9 to 6.

A rate is just a special type of ratio. A rate compares 2 quantities and, unlike a ratio, must have 2 different units. (here miles to hours. ) Units must be written. And you don’t say “to” to compare the numbers. You use “Per” when you have reduced the ratio.

A unit rate is how many units one quantity is per 1 unit of another quantity. There are some common unit rates you think about all the time. Earnings, driving, and comparing items in the grocery store is where you encounter unit rates.

Let’s calculate a unit rate. Look at the question. You should recognize this as a ratio problem because you see 2 quantities being compared, Now notice that the units are different, too, so you should recognize it as a rate problem. It’s rate when the questions asks about “per anything. Step 1, note the units in the order asked. Next, apply the numbers to those units .THen, divide the numerator by the denominator, and you get your unit rate of 80 mph.

Proportions are a relationship between 2 equal ratios that are always equal. So written like this ½ = 2/4. If the ratios aren’t equivalent, they aren’t proportional.

Understanding proportions requires NO new math, only an ability to work easily and automatically with equivalent fractions (So know your multiplication tables, too.)

To discover if ratios are proportional, you only have to cross multiply – all the magic happens in the cross-multiplying. Remember not to multiply across like in a fraction multiplication problem, I draw loops to make an X.

Multiply the numerator of the first ratio with denominator of the 2nd. That number goes on one side of the equal sign. Then multiply the remaining denominator with its opposite numerator, and that product should be equal to the first one. To check that 4/12 and 2/6 are proportional, cross multiply 4*6 and that should equal 2*12 – and it does? They are both 24. These two ratios are proportional. So here we have 3/9 is equal to 2/3 and we multiply the 3 (numerator) with the 3 (denominator) and then 9 (denominator) by 2 (numerator). These two products are not equal so this is not a proportional statement.

Cross multiply to find any missing quantity in a ratio problem. Here, multiply to get 24 = 12x. Now you only have to solve for x to see that x = 2. Now put that back into the problem and you can check with cross multiplication that 4/12 and 2/6 are equivalent. An important tip? No matter how complicated the numbers in the problem get, follow these steps religiously. You’ll never struggle.

If you can quickly and automatically recognize when a test question is a proportion problem, you’ll do well. Look for ratios (gallons, hours) Look for rate (per). So your denominator should end up as 1. If one quantity is missing, you have a proportion problem. And I know I use proportionality to solve for a missing quantity in ratios.

Use these steps to solve proportion problems. 1. Write the quantities being compared in the correct order . Here – gallons to miles. Write the numbers to match the quantities. 12 to 20. 2. Write the second ratio with X as the missing quantity (Remember, same order.) X gallons to 40 miles (doubled). 3. Cross multiply to set up your equation. 5. Solve for X.

You can solve all percent problems using proportions and starting with this formula. Each percent problem has a whole, a part, and a percent.(and a percent is always a number over 100). In this problem, you are given the part, the whole, and you always have the 100, so your % is the missing number. Insert the numbers in the right place in the formula, cross-multiply, and solve for x. By noticing the word What or How Many, you will understand which number is missing from the proportional relationship of ratios. Here 72* 100 = 600x. 7200 divided by 600 = this is the number over 100. It is 12, so 12% of the hats sold were summer hats.

Another percent problem – continue to look for the part, the whole, and the percent. Total hats sold is 600, and the percent is clearly given. So you have 600 / X = 12/100. Cross-multiply and solve for X. Here you will have 600 * 12 = 100x, so you have 7200 / 100 = 72. 72 hats were summer hats.. Check if that makes sense. Using your go to percent numbers, you know 10% would be 60, so its a bit more than that.

Third and last type of percent problem – and you should be noticing the format by now. Remember the formula? Good, because I recommend memorizing it. You immediately see the %, so 12 goes above the 100. You are looking for total, so X goes in the whole place, which leaves 72 to go in the part place. Cross multiply and solve for X. 7200 = 12X, 7200 divided by 12 = …600!

We’ve done it. See how there is no new math, it is only about reading the problems carefully and putting the numbers in the right place? If you will learn the difference between a ratio and a rate, get good at recognizing proportional relationships, or solving for them with cross-multiplication; learn to quickly recognize proportion problems and memorize the formula for solving percent problems. you can call yourself a “strong math student”.

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