Question

Unit 4

Unit ratios & proportions

Answer 1

[AI-generated answer — Bot GPT 3.5]

In mathematics, ratios and proportions are important concepts that are used to compare quantities and solve problems involving fractions.

A ratio is a comparison of two quantities or numbers. It is often written as "a:b" or "a to b." For example, if there are 10 boys and 20 girls in a class, the ratio of boys to girls is 10:20 or 1:2.

Proportions, on the other hand, are equations that state that two ratios are equal. If two ratios are proportional, it means that they have the same value. Proportions are often used to solve problems involving unknown quantities.

For example, suppose you know that the ratio of the length of a rectangle to its width is 3:2. If the width of the rectangle is 4, you can set up a proportion:

(length of rectangle) / (width of rectangle) = 3/2

Let x be the length of the rectangle. The proportion becomes:

x / 4 = 3/2

To solve for x, you can cross-multiply and solve for x:

2x = 12

x = 6

So the length of the rectangle is 6 units.

Ratios and proportions are used in various real-life situations such as cooking, finance, and geometry. They allow us to compare and understand relationships between quantities and make calculations based on those relationships.

Answer 2

[AI-generated answer — Step-by-Step Bot]

Step 1: Understanding Ratios

A ratio is a comparison of two quantities. It is expressed in the form "a:b" or "a/b", where "a" and "b" are numbers.

Step 2: Writing Ratios
To write a ratio, you need to identify two quantities that you want to compare and express them in the form "a:b". For example, if you want to compare the number of boys to the number of girls in a class of 20 students where there are 12 boys and 8 girls, you would write the ratio as 12:8 or 12/8.

Step 3: Simplifying Ratios
Sometimes, you can simplify ratios by dividing both numbers by their greatest common factor (GCF). For example, if you have the ratio 14:35, you can simplify it by dividing both numbers by 7 to get the simplified ratio 2:5.

Step 4: Proportions
A proportion is an equation that states that two ratios are equal. It is expressed in the form "a/b = c/d". For example, if you have the proportion 2/3 = 4/6, it means that the ratios 2:3 and 4:6 are equal.

Step 5: Solving Proportions
To solve a proportion, you can use cross-multiplication. Multiply the numerator of the first ratio by the denominator of the second ratio, and then multiply the denominator of the first ratio by the numerator of the second ratio. Set these two products equal to each other and solve for the unknown variable.

Step 6: Word Problems with Ratios and Proportions
In real-life situations, ratios and proportions are often used to solve word problems. To solve these problems, first identify the quantities being compared and set up a ratio or proportion. Then, use the techniques mentioned above to solve for the unknown variable.

That's it! These steps should help you understand and solve problems related to unit ratios and proportions in Unit 4.