What is a ratio?

A ratio is a relationship between two numbers that shows how many times the first number contains the second. It can be expressed in several forms: as a fraction, with a colon (e.g., 3:2), or with the word 'to' (e.g., 3 to 2).

How do you simplify a ratio?

To simplify a ratio, divide both terms of the ratio by their greatest common divisor (GCD). For example, to simplify the ratio 8:12, find the GCD, which is 4. Thus, 8 ÷ 4 : 12 ÷ 4 = 2:3.

What is a proportion?

A proportion is an equation that states that two ratios are equal. For example, if a/b = c/d, then a, b, c, and d are in proportion.

If 3/4 = x/20, what is the value of x?

Hint: Cross-multiply to solve for x.

Cross-multiplying gives 3 * 20 = 4 * x. Thus, 60 = 4x. Dividing both sides by 4 gives x = 15.

A recipe calls for a ratio of sugar to flour of 2:5. If you have 10 cups of flour, how much sugar do you need?

Hint: Set up a proportion.

Set up the proportion 2/5 = x/10. Cross-multiply: 2 * 10 = 5 * x, which simplifies to 20 = 5x. Dividing both sides by 5 gives x = 4 cups of sugar.

What does it mean if two ratios are equivalent?

If two ratios are equivalent, it means that they represent the same relationship between quantities. For example, 1:2 is equivalent to 2:4.

How can you express the ratio of 3 to 7 as a fraction?

The ratio of 3 to 7 can be expressed as the fraction 3/7.

If the ratio of boys to girls in a class is 5:3 and there are 40 boys, how many girls are there?

Hint: Use the ratio to find the total number of parts.

The total parts in the ratio 5:3 is 8. If there are 40 boys, each part represents 40/5 = 8. Therefore, girls = 3 * 8 = 24.

What is the formula for finding the part of a whole in a ratio?

If a:b = c, then a/(a+b) gives the fraction of the whole represented by a. For example, in the ratio 3:7, the part represented by 3 is 3/(3+7) = 3/10.

A map uses a scale of 1:100,000. If the distance on the map between two cities is 3 cm, what is the actual distance?

Hint: Multiply the map distance by the scale factor.

The actual distance is 3 cm * 100,000 = 300,000 cm or 3 km.

If a mixture contains 2 parts water to 3 parts alcohol, what is the ratio of alcohol to the total mixture?

Hint: First find the total parts.

The total parts are 2 + 3 = 5. The ratio of alcohol to the total mixture is 3:5.

In a survey, the ratio of dogs to cats is 4:3. If there are 28 dogs, how many cats are there?

Hint: Use the ratio to find the number of cats.

The ratio 4:3 means for every 4 dogs, there are 3 cats. If there are 28 dogs, then (28/4) * 3 = 21 cats.

What is the relationship between direct and inverse proportions?

In direct proportion, as one quantity increases, the other also increases; in inverse proportion, as one quantity increases, the other decreases. For example, if y is directly proportional to x, then y = kx; if inversely, then y = k/x.

If the ratio of the ages of two siblings is 5:7 and the older sibling is 21 years old, how old is the younger sibling?

Hint: Set up a proportion based on the ratio.

If the ratio is 5:7, then the younger sibling's age can be found by setting up the proportion 5/7 = x/21. Cross-multiplying gives 5 * 21 = 7x, so x = 15.

If 1:3 is the ratio of apples to oranges, how many oranges are there if there are 12 apples?

Hint: Use the ratio to find the total parts.

The total parts in the ratio 1:3 is 4. If there are 12 apples, then each part is 12/1 = 12. Therefore, oranges = 3 * 12 = 36.