Ratios and Proportional Relationships
What is a ratio?
A ratio is a comparison of two or more numbers. It can be written in the form of a fraction, using a colon, or using the word "to".
What is a rational number?
A rational number is any number that can be written as a fraction, where the numerator and denominator are both integers.
What is the distributive property?
The distributive property states that when you multiply a number by a sum or difference, you can multiply each term inside the parentheses by that number separately. For example, $a(b + c) = ab + ac$.
Define acute angle.
An acute angle is an angle that measures less than 90 degrees.
What is random sampling?
Random sampling is a method of selecting a sample from a population in such a way that every member of the population has an equal chance of being selected.
Solve the proportion: $\frac{4}{x}$ = $\frac{6}{9}$.
Cross-multiplying, we get $4 \times 9 = 6 \times x$. Simplifying, we find $36 = 6x$. Dividing both sides by 6, we get $x = 6$.
Evaluate: $\frac{5}{8} + \frac{3}{4}$.
To add fractions, we need a common denominator. In this case, the common denominator is 8. Adding the fractions, we get $\frac{5}{8} + \frac{6}{8} = \frac{11}{8}$. This can also be written as $1 \frac{3}{8}$.
Simplify: $2(x - 3) - 4(x + 1)$.
Using the distributive property, we can simplify the expression: $2x - 6 - 4x - 4$. Combining like terms, we get $-2x - 10$.
Calculate the area of a rectangle with length 8 cm and width 5 cm.
The area of a rectangle is found by multiplying the length by the width. In this case, the area is $8 \times 5 = 40$ square cm.
A survey of 100 students found that 60% of them preferred pizza over hamburgers. How many students preferred pizza?
To find the number of students who preferred pizza, we multiply the percentage by the total number of students: $0.6 \times 100 = 60$. Therefore, 60 students preferred pizza.
A recipe calls for 3 cups of flour and makes 24 cookies. How many cookies can be made with 5 cups of flour?
We can set up a proportion: $\frac{3}{24} = \frac{5}{x}$. Cross-multiplying, we get $3x = 120$. Dividing both sides by 3, we find $x = 40$. Therefore, 40 cookies can be made with 5 cups of flour.
Simplify: $2 \frac{1}{2} + 3 \frac{1}{3}$.
To simplify mixed numbers, we convert them to improper fractions. $2 \frac{1}{2}$ is equivalent to $\frac{5}{2}$, and $3 \frac{1}{3}$ is equivalent to $\frac{10}{3}$. Adding the fractions, we get $\frac{5}{2} + \frac{10}{3}$. To add fractions with different denominators, we need a common denominator. In this case, the common denominator is 6. Adding the fractions, we get $\frac{15}{6}$. Simplifying, we find $2 \frac{1}{2} + 3 \frac{1}{3} = 2 \frac{1}{2}$.
Solve the equation: $2(3x - 5) = 10$.
Distributing the 2 on the left side, we get $6x - 10 = 10$. Adding 10 to both sides, we have $6x = 20$. Dividing both sides by 6, we find $x = \frac{10}{3}$.
Determine the surface area of a cube with side length 2 cm.
The surface area of a cube is found by multiplying the length of one side by itself, and then multiplying by 6 (since a cube has 6 equal faces). In this case, the surface area is $2 \times 2 \times 6 = 24$ square cm.
A bag contains 4 red marbles, 6 blue marbles, and 2 green marbles. What is the probability of drawing a red marble?
The probability of drawing a red marble is the number of red marbles divided by the total number of marbles: $\frac{4}{12} = \frac{1}{3}$.
The ratio of red cars to blue cars in a parking lot is 3:5. If there are 36 red cars, how many blue cars are there?
We can set up a proportion: $\frac{3}{5} = \frac{36}{x}$. Cross-multiplying, we get $3x = 180$. Dividing both sides by 3, we find $x = 60$. Therefore, there are 60 blue cars.
Solve: $1.2 \times (-\frac{3}{4})$.
Multiplying decimals and fractions, we get $1.2 \times (-\frac{3}{4}) = -0.9$.
Solve the equation: $4y - 2(2y + 3) = 5$.
Distributing the -2 on the left side, we get $4y - 4y - 6 = 5$. Combining like terms, we have $-6 = 5$. Since this equation is not true, there is no solution.
Find the volume of a cylinder with radius 4 cm and height 10 cm. Use $\pi$ as 3.14.
The volume of a cylinder is found by multiplying the area of the base (which is the area of a circle with radius $r$) by the height. In this case, the volume is $3.14 \times 4^2 \times 10 = 502.4$ cubic cm.
What is the probability of rolling an even number on a standard six-sided die?
There are three even numbers (2, 4, and 6) out of a total of six possible outcomes. So, the probability of rolling an even number is $\frac{3}{6} = \frac{1}{2}$.
A car travels 150 miles in 3 hours. How far will it travel in 7.5 hours?
We can set up a proportion: $\frac{150}{3} = \frac{x}{7.5}$. Cross-multiplying, we get $3x = 1125$. Dividing both sides by 3, we find $x = 375$. Therefore, the car will travel 375 miles in 7.5 hours.
Mrs. Smith spent $\frac{5}{8}$ of her budget on books and movies. If she spent \$32 on books, what is her total budget?
We can set up a proportion: $\frac{5}{8} = \frac{32}{x}$. Cross-multiplying, we get $5x = 256$. Dividing both sides by 5, we find $x = 51.2$. Therefore, Mrs. Smith's total budget is \$51.20.
A triangle has a perimeter of 25 cm. If one side measures 6 cm and another side measures 9 cm, what is the length of the third side?
Let the length of the third side be $x$. Using the perimeter formula, we can set up the equation $6 + 9 + x = 25$. Simplifying, we find $x = 10$. Therefore, the length of the third side is 10 cm.
Two angles in a triangle measure 30° and 70°. What is the measure of the third angle?
The sum of the angles in a triangle is always 180 degrees. So, to find the measure of the third angle, we subtract the measures of the two given angles from 180: $180 - 30 - 70 = 80$ degrees. Therefore, the measure of the third angle is 80°.
If two fair coins are flipped, what is the probability of getting at least one head?
To find the probability of getting at least one head, we need to find the probability of getting no heads and subtract it from 1. The probability of getting no heads is $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. So, the probability of getting at least one head is $1 - \frac{1}{4} = \frac{3}{4}$.