Ratios can be used to show the relation between any quantities, even if one is not directly tied to the other (as they would be in a recipe). For example, if there are five girls and ten boys in a class, the ratio of girls to boys is 5 to 10. Neither quantity is dependent on or tied to the other, and would change if anyone left or new students came in. The ratio merely compares the quantities.
You will commonly see ratios represented using words (as above). Because they are used so commonly and in such a variety of ways, if you find yourself working outside of mathematic or scientific fields, this may the most common form of ratio you will see. Ratios are frequently expressed using a colon. When comparing two numbers in a ratio, you’ll use one colon (as in 7 : 13). When you’re comparing more than two numbers, you’ll put a colon between each set of numbers in succession (as in 10 : 2 : 23). In our classroom example, we might compare the number of boys to the number of girls with the ratio 5 girls : 10 boys. We can simply express the ratio as 5 : 10. Ratios are also sometimes expressed using fractional notation. In the case of the classroom, the 5 girls and 10 boys would be shown simply as 5/10. That said, it shouldn’t be read out loud the same as a fraction, and you need to keep in mind that the numbers do not represent a portion of a whole.
In the classroom example above, 5 girls to 10 boys (5 : 10), both sides of the ratio have a factor of 5. Divide both sides by 5 (the greatest common factor) to get 1 girl to 2 boys (or 1 : 2). However, we should keep the original quantities in mind, even when using this reduced ratio. There are not 3 total students in the class, but 15. The reduced ratio just compares the relationship between the number of boys and girls. There are 2 boys for every girl, not exactly 2 boys and 1 girl. Some ratios cannot be reduced. For example, 3 : 56 cannot be reduced because the two numbers share no common factors - 3 is a prime number, and 56 is not divisible by 3.
For example, a baker needs to triple the size of a cake recipe. If the normal ratio of flour to sugar is 2 to 1 (2 : 1), then both numbers must be increased by a factor of three. The appropriate quantities for the recipe are now 6 cups of flour to 3 cups of sugar (6 : 3). The same process can be reversed. If the baker needed only one-half of the normal recipe, both quantities could be multiplied by 1/2 (or divided by two). The result would be 1 cup of flour to 1/2 (0. 5) cup of sugar.
For example, let’s say we have a small group of students containing 2 boys and 5 girls. If we were to maintain this proportion of boys to girls, how many boys would be in a class that contained 20 girls? To solve, first, let’s make two ratios, one with our unknown variables: 2 boys : 5 girls = x boys : 20 girls. If we convert these ratios to their fraction forms, we get 2/5 and x/20. If you cross multiply, you are left with 5x=40, and you can solve by dividing both figures by 5. The final solution is x=8.
Wrong method: “8 - 4 = 4, so I added 4 potatoes to the recipe. That means I should take the 5 carrots and add 4 to that too. . . wait! That’s not how ratios work. I’ll try again. " Right method: “8 ÷ 4 = 2, so I multiplied the number of potatoes by 2. That means I should multiply the 5 carrots by 2 as well. 5 x 2 = 10, so I want 10 carrots total in the new recipe. "
A dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in the dragon’s hoard? Grams and kilograms are not the same unit, so we’ll need to convert. 1 kilogram = 1,000 grams, so 10 kilograms = 10 kilograms x 1,000grams1kilogram{\displaystyle {\frac {1,000grams}{1kilogram}}} = 10 x 1,000 grams = 10,000 grams. The dragon has 500 grams of gold and 10,000 grams of silver. The ratio of gold to silver is 500gramsGold10,000gramsSilver=5100=120{\displaystyle {\frac {500gramsGold}{10,000gramsSilver}}={\frac {5}{100}}={\frac {1}{20}}}.
Example problem: If you have six boxes, and in every three boxes there are nine marbles, how many marbles do you have? Wrong method: 6boxes∗3boxes9marbles=. . . {\displaystyle 6boxes*{\frac {3boxes}{9marbles}}=. . . } Wait, nothing cancels out, so my answer would be “boxes x boxes / marbles. " That doesn’t make sense. Right method:6boxes∗9marbles3boxes={\displaystyle 6boxes*{\frac {9marbles}{3boxes}}=} 6boxes∗3marbles1box={\displaystyle 6boxes*{\frac {3marbles}{1box}}=}6boxes∗3marbles1box={\displaystyle {\frac {6boxes3marbles}{1box}}=} 6∗3marbles1={\displaystyle {\frac {63marbles}{1}}=} 18 marbles.
