Content

• Steps
• Part 1 of 3: Determining Relationships
• Part 2 of 3: Using ratios
• Part 3 of 3: Common mistakes

A ratio (in mathematics) is a relationship between two or more numbers of the same kind. Ratios compare absolute values ​​or parts of a whole. Ratios are calculated and written in different ways, but the basic principles are the same for all ratios.

Steps

Part 1 of 3: Determining Relationships

1. 1 Using ratios. Ratios are used both in science and in everyday life to compare values. The simplest ratios relate only two numbers, but there are ratios that compare three or more values. In any situation in which more than one quantity is present, a ratio can be written. By linking some values, ratios can, for example, suggest how to increase the amount of ingredients in a recipe or substances in a chemical reaction.
2. 2 Determination of ratios. A ratio is a relationship between two (or more) values ​​of the same kind. For example, if you need 2 cups of flour and 1 cup of sugar to make a cake, then the ratio of flour to sugar is 2 to 1.
   • The ratios can also be used in cases where the two quantities are not related to each other (as in the example with the cake). For example, if there are 5 girls and 10 boys in a class, then the ratio of girls to boys is 5 to 10. These values ​​(the number of boys and the number of girls) are independent of each other, that is, their values ​​will change if someone leaves the class or a new student will come to the class. Ratios simply compare the values ​​of quantities.
3. 3 Pay attention to the different ways of representing ratios. Relationships can be expressed in words or using mathematical symbols.
   • Very often the ratios are expressed in words (as shown above). Especially this form of representation of ratios is used in everyday life, far from science.
   • Also, ratios can be expressed through a colon. When comparing two numbers in a ratio, you will use one colon (for example, 7:13); when comparing three or more values, put a colon between each pair of numbers (for example, 10: 2: 23). In our class example, you can express the ratio of girls to boys like this: 5 girls: 10 boys. Or like this: 5:10.
   • Less commonly, ratios are expressed using a slash. In the class example, it can be written like this: 5/10. Nevertheless, this is not a fraction and such a ratio is not read as a fraction; Moreover, remember that in the ratio, the numbers do not represent part of a whole.

Part 2 of 3: Using ratios

1. 1 Simplify the ratio. The ratio can be simplified (similar to fractions) by dividing each term (number) of the ratio by the greatest common factor. However, do not lose sight of the original ratio values ​​when doing this.
   • In our example, there are 5 girls and 10 boys in the class; the ratio is 5:10. The greatest common divisor of the terms of the ratio is 5 (since both 5 and 10 are divisible by 5). Divide each ratio number by 5 to get the ratio of 1 girl to 2 boys (or 1: 2). However, keep the original values ​​in mind when simplifying the ratio. In our example, there are not 3 students in the class, but 15. The simplified ratio compares the number of boys and the number of girls. That is, for every girl there are 2 boys, but there are not 2 boys and 1 girl in the class.
   • Some relationships are not simplified. For example, the ratio 3:56 is not simplified because these numbers have no common divisors (3 is a prime number, and 56 is not divisible by 3).
2. 2 Use multiplication or division to increase or decrease the ratio. Common tasks in which it is necessary to increase or decrease two values ​​proportional to each other. If you are given a ratio and need to find the corresponding greater or lesser ratio, multiply or divide the original ratio by some given number.
   • For example, a baker needs to triple the amount of ingredients given in a recipe. If the recipe has a flour to sugar ratio of 2 to 1 (2: 1), then the baker will multiply each term in the ratio by 3 to get a 6: 3 ratio (6 cups flour to 3 cups sugar).
   • On the other hand, if the baker needs to halve the amount of ingredients given in the recipe, then the baker will divide each term in the ratio by 2 and get a ratio of 1: ½ (1 cup flour to 1/2 cup sugar).
3. 3 Finding an unknown value when two equivalent relationships are given. This is a problem in which you need to find an unknown variable in one relation using the second relation, which is equivalent to the first. To solve such problems, use criss-cross multiplication. Write down each ratio as an ordinary fraction, put an equal sign between them and multiply their terms crosswise.
   • For example, a group of students is given, in which there are 2 boys and 5 girls. What will be the number of boys if the number of girls is increased to 20 (the proportion remains the same)? First, write down two ratios - 2 boys: 5 girls and NS boys: 20 girls. Now write these ratios as fractions: 2/5 and x / 20. Multiply the terms of the fractions crosswise to get 5x = 40; therefore, x = 40/5 = 8.

Part 3 of 3: Common mistakes

1. 1 Avoid addition and subtraction in ratio word problems. Many word problems look something like this: “In the recipe, you need to use 4 potato tubers and 5 carrot roots. If you want to add 8 potato tubers, how many carrots do you need to keep the ratio unchanged? " When solving such problems, students often make the mistake of adding the same amount of ingredients to the original number. However, to keep the ratio, you need to use multiplication.Here are examples of right and wrong solutions:
   • False: “8 - 4 = 4 - so we added 4 potato tubers. So, you need to take 5 carrot root crops and add 4 more to them ... Stop! Relationships are not calculated that way. It is worth trying again. "
   • It is true: "8 ÷ 4 = 2 - so we multiplied the amount of potatoes by 2. Accordingly, 5 carrots must also be multiplied by 2. 5 x 2 = 10 - 10 carrots must be added to the recipe."
2. 2 Convert terms to the same units. Some word problems are made more difficult by adding different units of measurement. Convert them before calculating the ratio. Here's an example of a problem and solution:
   • The dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in the dragon's treasury?
   • Grams and kilograms are different units of measure, they need to be converted. 1 kilogram = 1000 grams, respectively, 10 kilograms = 10 kilograms x 1000 grams / 1 kilogram = 10 x 1000 grams = 10,000 grams.
   • The dragon has 500 grams of gold and 10,000 grams of silver in its treasury.
   • The ratio of gold to silver is: 500 grams of gold / 10,000 grams of silver = 5/100 = 1/20.
3. 3 Write down the units of measurement after each value. In word problems, it is much easier to recognize an error if you write down the units after each value. Remember that quantities with the same unit in both the numerator and denominator are canceled. By shortening the expression, you get the right answer.
   • Example: 6 boxes are given, in every third box there are 9 balls. How many balls are there?
   • Incorrect: 6 boxes x 3 boxes / 9 balls = ... Stop, nothing can be cut. The answer would be "boxes x boxes / balls". It doesn't make sense.
   • Correct: 6 boxes x 9 balls / 3 boxes = 6 boxes * 3 balls / 1 box = 6 boxes * 3 balls / 1 box = 6 * 3 balls / 1 = 18 balls.