Foundations of Ratios and Proportions

Foundational

Learn the fundamental concepts of ratios, including how to write, simplify, and interpret them. This stack also introduces proportions as statements of equality between two ratios, forming the basis for more complex problem-solving.

Foundations of Ratios and Proportions

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What is a ratio?

A ratio is a way to show a relationship or comparison between two or more quantities of the same unit. It indicates how many times one number contains another.

What are the three common ways to write a ratio between two quantities, $a$ and $b$?

1. Using the word 'to': $a$ to $b$ 2. Using a colon: $a:b$ 3. As a fraction: $\frac{a}{b}$ (where $b \neq 0$)

A class has $15$ boys and $12$ girls. What is the ratio of boys to girls, expressed in its simplest form?

The initial ratio is $15:12$. To simplify, find the greatest common divisor of $15$ and $12$, which is $3$. Divide both parts by $3$: $15 \div 3 = 5$ and $12 \div 3 = 4$. The simplified ratio is $5:4$.

How does a 'part-to-part' ratio differ from a 'part-to-whole' ratio?

A part-to-part ratio compares two distinct groups (e.g., cats to dogs). A part-to-whole ratio compares one group to the total (e.g., cats to all animals). For a group of $3$ cats and $5$ dogs, the part-to-part ratio of cats to dogs is $3:5$, and the part-to-whole ratio of cats to total pets is $3:8$.

What does a three-part ratio like $a:b:c$ represent?

It compares three quantities simultaneously. For example, a recipe might require flour, sugar, and butter in a ratio of $3:2:1$. This means for every $3$ cups of flour, you need $2$ cups of sugar and $1$ cup of butter.

What is a proportion?

A proportion is an equation stating that two ratios are equivalent. It is written in the form $\frac{a}{b} = \frac{c}{d}$.

How can you determine if two ratios form a valid proportion?

Two ratios, $\frac{a}{b}$ and $\frac{c}{d}$, form a proportion if their cross-products are equal. That is, if $a \times d = b \times c$. This is also known as the means-extremes property.

Set up a proportion for this problem: If $3$ apples cost $\$2$, how much will $15$ apples cost?

Let $x$ be the unknown cost. The proportion can be set up by matching the units in the ratios: $\frac{\text{apples}}{\text{cost}}$. $$ \frac{3 \text{ apples}}{\$2} = \frac{15 \text{ apples}}{x} $$

What is the key difference between a ratio and a rate?

A ratio compares quantities with the same units (e.g., boys to girls). A rate compares quantities with different units (e.g., miles per hour, or dollars per pound). The units are an essential part of a rate.

The ratio of width to height for a rectangle is $7:4$. If the width is $21$ cm, what is the height?

Set up a proportion: $\frac{\text{width}}{\text{height}} = \frac{7}{4} = \frac{21}{h}$. By noticing that $7 \times 3 = 21$, you can find $h$ by multiplying $4$ by $3$. The height is $12$ cm.

In a bag of marbles, the ratio of red, green, and blue marbles is $2:5:3$. If there are $40$ marbles in total, how many green marbles are there?

The sum of the ratio parts is $2+5+3 = 10$. This means the green marbles make up $\frac{5}{10}$ of the total. Calculate the number of green marbles: $\frac{5}{10} \times 40 = 20$. There are $20$ green marbles.

Foundations of Ratios and Proportions

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