ACT math proportions questions

What is a proportion?

A statement that two ratios are equal, expressed as $\displaystyle \frac{a}{b} = \frac{c}{d}$.

True or false: Proportions can only compare whole numbers.

False, because proportions can involve fractions and decimals.

Ratio of 1:4 means...

For every 1 part of a substance, there are 4 parts of another.

Fill in the blank: $\displaystyle \frac{3}{x} = \frac{6}{12}$

Solve for x: $\displaystyle x = 6$.

Difference between direct and inverse proportions

Direct: as one increases, the other increases. Inverse: as one increases, the other decreases.

If 3 apples cost $2, how much do 6 apples cost?

Using proportions: $\displaystyle \frac{3}{2} = \frac{6}{x} ightarrow x = 4$.

What is cross-multiplication?

A method to solve proportions: if $\displaystyle \frac{a}{b} = \frac{c}{d}$, then $\displaystyle ad = bc$.

True or false: Ratios can be expressed as fractions.

True, because a ratio is essentially a fraction.

If a car travels 60 miles in 1 hour, how far in 2 hours?

Using proportions: $\displaystyle 60 = \frac{distance}{2} ightarrow distance = 120$ miles.

When is a proportion considered equivalent?

When the products of the means and extremes are equal.

What does the term 'scale factor' mean?

The ratio of the dimensions of a model to the actual object.

If 5 pencils cost $1, how much for 15 pencils?

$\displaystyle 1 imes \frac{15}{5} =$3.

True or false: You can only set up one proportion for a problem.

False, because multiple proportions can be set for complex problems.

If the ratio of boys to girls is 2:3 and there are 12 girls, how many boys?

Let boys = x: $\displaystyle x/12 = 2/3 ightarrow x = 8$.

If 4 cups of flour make 12 cookies, how much for 10 cookies?

$\displaystyle \frac{4}{12} = \frac{x}{10} ightarrow x = \frac{10 imes 4}{12} = \frac{40}{12} ightarrow x ext{ approx } 3.33$.

Difference between unit rate and ratio

Unit rate is a ratio comparing quantities with a denominator of 1.

If a map scale is 1:100, what does this mean?

1 unit on the map represents 100 units in reality.

If it takes 5 hours to paint 3 walls, how long for 9 walls?

Set up the proportion: $\displaystyle 5/3 = x/9 ightarrow x = 15$ hours.

What is a unit rate?

A rate with a denominator of 1, showing the amount per one unit.

If 8 oranges cost $4, what is the cost of 20 oranges?

$\displaystyle 4 imes \frac{20}{8} =$10.

If a recipe requires 2 cups of sugar for 3 servings, how much for 6 servings?

Set up: $\displaystyle \frac{2}{3} = \frac{x}{6} ightarrow x = 4$ cups.

If the ratio of cats to dogs is 1:4, how many dogs if there are 5 cats?

Using ratios: $\displaystyle 5 imes 4 = 20$ dogs.

If 7 miles cost $14, what is the cost per mile?

$\displaystyle 14/7 =$2 per mile.