Proportions Flashcards
What is the formula for basic proportion?
a/b = c/d
In a basic proportion scenario, how do you set up the equation if 3 apples cost $2 and you want to find the cost of 9 apples?
3/2 = 9/x
Set up a proportion: A proportion is an equation stating that two ratios are equal. In this case, we’ll set up a proportion comparing the number of apples to their cost:
Apples / Cost = Apples / Cost
Plug in the known values: We know that 3 apples cost $2, and we want to find the cost of 9 apples. Let ‘x’ represent the unknown cost:
3 / $2 = 9 / x
Cross-multiply: Multiply the numerator of one fraction by the denominator of the other, and set the two products equal to each other:
3 * x = 2 * 9
Simplify:
3x = 18
Solve for x: Divide both sides of the equation by 3 to isolate x:
x = 18 / 3
Calculate:
x = $6
What is the solution for x when solving the equation 3x = 18?
x = $6
What is the formula for scale drawings/maps?
A map has a scale of 1 centimeter = 25 kilometers. If two cities are 4 centimeters apart on the map, what is the actual distance between them?
Map Distance / Actual Distance = Map Scale
If 1 cm = 25 km, then 4 cm represents 4 * 25 km = 100 km.
Set up the proportion:
1 cm / 25 km = 4 cm / x km (where x is the actual distance)
Cross-multiply:
1 * x = 25 * 4
Solve for x:
x = 100
If a map has a scale of 1 inch = 50 miles, how do you find the actual distance between two cities that are 3 inches apart?
3 inches / x miles = 1 inch / 50 miles
Let ‘M’ represent the Map Distance.
Let ‘A’ represent the Actual Distance.
Let ‘m’ represent the Map Scale Unit (in this case, 1 inch).
Let ‘a’ represent the Actual Scale Unit (in this case, 50 miles).
The formula becomes:
M / A = m / a
What is the actual distance between two cities that are 3 inches apart on a map with a scale of 1 inch = 50 miles?
(Map Distance) / (Actual Distance) = (Map Scale Unit) / (Actual Scale Unit)
3/Actual DIstance=1/50 then cross multiply
150miles
What is the formula that relates the sides of similar figures?
Two triangles are similar. The smaller triangle has sides of 4 and 6. The larger triangle has a corresponding side of 12. What is the length of the other corresponding side of the larger triangle?
a/b = c/d
4/12 = 6/x
4x = 72
x = 18
If two triangles are similar and the smaller triangle has sides of 3, 4, and 5, how do you find the shortest side of the larger triangle if its longest side is 10?
Imagine you have a picture of a triangle. Then you use a copy machine to make a bigger copy of it. The new triangle is similar to the old one. They look exactly the same, just one is bigger.
Now, let’s say the small triangle has sides of 3, 4, and 5 inches. The longest side is 5 inches.
You make a bigger copy, and now the longest side is 10 inches.
To find how much bigger the triangle got, we can compare the longest sides:
The small triangle’s longest side is 5.
The big triangle’s longest side is 10.
10 is twice as big as 5 (because 5 + 5 = 10, or 5 x 2 = 10). So, the big triangle is twice the size of the small triangle.
That means all the sides of the big triangle are twice as big as the sides of the small triangle.
The shortest side of the small triangle is 3. So, to find the shortest side of the big triangle, we just double it:
3 x 2 = 6
So, the shortest side of the larger triangle is 6 inches.
What is the solution for x when solving the equation 5x = 30?
x = 6
What is the formula for rate?
Rate = Quantity / Time
Speed/Velocity: Rate = Distance / Time
Work Rate: Rate = Amount of Work / Time
Consumption Rate: Rate = Amount Consumed / Time
If a car travels 120 miles in 2 hours, how do you find how far it will travel in 5 hours?
Step 1: Calculate the car’s speed.
Speed is calculated by dividing the distance traveled by the time taken.
Formula: Speed (v) = Distance (d) / Time (t)
Calculation: v = 120 miles / 2 hours = 60 miles per hour
Step 2: Calculate the new distance traveled.
Now that we know the speed, we can calculate the distance the car travels in a different amount of time. Distance is calculated by multiplying speed by time.
Formula: Distance (d) = Speed (v) * Time (t)
Calculation: d = 60 miles per hour * 5 hours = 300 miles
What is the distance a car will travel in 5 hours if it travels 120 miles in 2 hours?
Step 1:
Speed=distance/time
speed=120/2
speed=60m/ph
Step 2:
Distance=speed x time
Distance = 60m/ph x 5hrs
Distance = 300miles
Calculate the speed of the car, then use the speed to calculate the distance traveled in 5hrs
What is the formula for direct variation?
The cost of gasoline is directly proportional to the number of gallons purchased. If 3 gallons of gas cost $9, how much will 7 gallons cost?
y = kx (k = constant of variation)
y is the dependent variable.
x is the independent variable.
k is the constant of variation (or constant of proportionality).
This constant represents the ratio between y and x
Find the constant of variation (k):
We know that y (cost) = $9 when x (gallons) = 3.
Using the formula y = kx:
$9 = k * 3
k = $9 / 3
k = $3/gallon (This means each gallon costs $3)
Use the constant of variation to find the cost of 7 gallons:
Now we know k = $3/gallon and x = 7 gallons.
Using the formula y = kx:
y = $3/gallon * 7 gallons
y = $21
Therefore, 7 gallons of gas will cost $21.
If the cost of gas is directly proportional to gallons purchased, how do you find the cost for 7 gallons if 3 gallons cost $9?
$9 = k * 3; k = $3/gallon; Cost = $3/gallon * 7 gallons
Independent Variable and Dependent Variable
In Math (like y = kx):
x (Independent): This is the value you choose or control.
y (Dependent): This is the value that changes because of the x value.
A constant of variation (k) is a ratio that represents the relationship between the independent variable (x) and the dependent variable (y). It can be found by dividing y by x, if both of those variables have known values.
What is the formula for basic rate?
Rate = Distance / Time
If a train travels 240 miles in 4 hours, what is its average speed?
Rate = 60 miles/hour
What is the formula for work rate?
Work = Rate * Time
How much of a room can John paint in 2 hours if he can paint the whole room in 6 hours?
Work = 1/3 of the room
What is the scenario for combined work rate?
Two or more people working together
If Maria can type 60 words per minute and David can type 40 words per minute, how many words can they type together in 5 minutes?
Total Words = 500 words
What is the formula for basic ratio?
Ratio = Part A / Part B (or A:B)
If the ratio of boys to girls in a class is 3:2 and there are 12 boys, how many girls are there?
x = 8 girls
What is the scenario for ratio with total?
Ratio given with a total quantity
In a fruit basket with a ratio of apples to oranges of 2:3 and 20 fruits total, how many are oranges?
Find the total parts of the ratio: The ratio of apples to oranges is 2:3. This means there are 2 + 3 = 5 parts in total.
Determine the value of one part: There are 20 fruits in total, and they are divided into 5 parts. So, one part represents 20 fruits / 5 parts = 4 fruits.
Calculate the number of oranges: The ratio shows that there are 3 parts oranges. Since one part represents 4 fruits, there are 3 parts * 4 fruits/part = 12 oranges.
So, there are 12 oranges in the fruit basket.
What is the scenario for part-to-whole ratio?
Comparing a part to the whole
If there are 8 red marbles and 12 blue marbles, what is the ratio of red marbles to the total number of marbles?
8/20 (which can be simplified to 2/5)
What is the definition of the Greatest Common Divisor (GCD)?
The largest number that divides evenly into two or more numbers
What is GCD?
Imagine you have two bags of candy:
One bag has 12 candies.
The other bag has 18 candies.
You want to share the candies equally with your friends. What’s the biggest number of friends you can share with so that everyone gets the same number of candies from both bags?
That’s where the Greatest Common Divisor (GCD) comes in. It’s the biggest number that divides evenly into both 12 and 18.
Here’s how to find it:
List the factors (numbers that divide evenly) of each number:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Find the common factors (numbers that are on both lists):
Common factors of 12 and 18: 1, 2, 3, 6
Find the greatest (biggest) common factor:
The biggest number on the common factors list is 6.
So, the Greatest Common Divisor (GCD) of 12 and 18 is 6. This means you can share the candies with 6 friends, and each friend will get 2 candies from the first bag (12 / 6 = 2) and 3 candies from the second bag (18 / 6 = 3).
Another Example:
You have two pieces of rope:
One is 8 meters long.
The other is 12 meters long.
You want to cut the rope into equal pieces, and you want the pieces to be as long as possible. What’s the longest length you can cut the pieces?
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Common factors: 1, 2, 4
Greatest common factor: 4
So, you can cut each rope into pieces that are 4 meters long.
In short: The Greatest Common Divisor is the biggest number that can divide evenly into two (or more) numbers. It helps you find the largest equal groups you can make from two different amounts.
What is the definition of the Least Common Multiple (LCM)?
The smallest number that is a multiple of two or more numbers
What is LCM
Imagine you’re trying to compare pieces of pizza. But the pizzas are cut into different numbers of slices!
One pizza is cut into 2 slices.
Another pizza is cut into 4 slices.
To easily compare how much pizza you have, you want to cut the slices so they’re the same size. That’s what the “lowest common denominator” helps you do.
The “denominator” is the bottom number of a fraction. It tells you how many total slices the pizza is cut into.
The “lowest common denominator” is the smallest number that both denominators can divide into evenly. It’s like finding a way to cut both pizzas so they have the same number of slices, and that number is the smallest possible.
In our pizza example:
The first pizza has a denominator of 2.
The second pizza has a denominator of 4.
The lowest common denominator is 4. Why? Because you can easily cut the first pizza (with 2 slices) into 4 slices by cutting each slice in half. The second pizza already has 4 slices.
Another Example:
You have two fractions: 1/3 and 1/6.
The denominators are 3 and 6.
The lowest common denominator is 6. Why? Because 6 can be divided by both 3 (3 x 2 = 6) and 6 (6 x 1 = 6).
In short: The lowest common denominator is the smallest number that all the bottom numbers of your fractions can divide into evenly.
It helps you compare and add or subtract fractions more easily.
Sources and related content
What is a prime number?
A whole number greater than 1 that has only two divisors: 1 and itself
Is 7 a prime number?
Yes, 7 is prime
What is a composite number?
A composite number is a whole number that has more than two factors (divisors). In simpler terms, it’s a number that can be divided evenly by numbers other than 1 and itself.
Here’s a breakdown:
Factors: Factors are numbers that divide evenly into another number. For example, the factors of 6 are 1, 2, 3, and 6 because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1.
Prime Numbers: Prime numbers have only two factors: 1 and themselves. For example, 7 is a prime number because its only factors are 1 and 7.
Composite Numbers: Composite numbers have more than two factors. For example, 6 is a composite number because it has four factors (1, 2, 3, and 6).
Examples of Composite Numbers:
4 (factors: 1, 2, 4)
6 (factors: 1, 2, 3, 6)
8 (factors: 1, 2, 4, 8)
9 (factors: 1, 3, 9)
10 (factors: 1, 2, 5, 10)
12 (factors: 1, 2, 3, 4, 6, 12)
Important Note: The number 1 is neither prime nor composite. It only has one factor (itself).
Sources and related content
Is 9 a composite number?
Yes, 9 is composite
What is the definition of an integer?
All whole numbers (positive, negative, and zero)
Give some examples of integers.
-3, -2, -1, 0, 1, 2, 3
What is the definition of a rational number?
In short: If you can write a number as a fraction of two whole numbers (with the bottom number not being zero), it’s a rational number.
Examples of Rational Numbers:
Fractions: 1/2, 3/4, -5/8, 10/3
Integers: 5 (can be written as 5/1), -3 (can be written as -3/1), 0 (can be written as 0/1)
Decimals that terminate (end): 0.25 (can be written as 1/4), 1.5 (can be written as 3/2)
Decimals that repeat: 0.333… (can be written as 1/3), 0.142857142857… (can be written as 1/7)
Its not rational when it repeats forever like pie
Are 0.5 and 3/4 rational numbers?
Yes
What is an algebraic expression?
A combination of variables, constants, and operations (addition, subtraction, multiplication, division) without an equals sign
Provide an example of an algebraic expression for: A candy store sells two types of candy: lollipops and chocolates. Lollipops cost $2 each, and chocolates cost $3 each. You buy some lollipops and some chocolates. However, you have a coupon for $5 off your total purchase. Write an expression to represent your final cost.
2x + 3y - 5
2x represents the total cost of lollipops (2 dollars each).
3y represents the total cost of chocolates (3 dollars each).
-5 represents a $5 discount.
If you bought 3 lollipops (x = 3) and 4 chocolates (y = 4), you could use the expression to find your total cost:
2(3) + 3(4) - 5 = 6 + 12 - 5 = $13
What is the definition of a polynomial?
Imagine you have building blocks. Some are just plain blocks (like the number 5). Some have a special sticker on them that we’ll call “x” (like x or 2x or x times x).
A polynomial is like a building made with these blocks, using only these rules:
You can put blocks together by adding them (like 5 + x).
You can make blocks bigger by multiplying them by numbers (like 2 times x, which we write as 2x).
You can even multiply the “x” block by itself (like x times x, which we write as x²).
But you cannot do these things:
You can’t divide by “x” (like 5 divided by x).
You can’t have “x” under a square root sign (like the square root of x).1
Provide an example of a polynomial.
x² + 4x - 7
What is the definition of an equation?
A mathematical statement that two expressions are equal, having an equals sign
Solve for x: x + 7 = 15.
x = 8
What are some common math vocabulary terms?
Sum, Difference, Product, Quotient, Is/Equals, More than, Less than
What is the formula to find a new value after a percentage increase?
New Value = Original Value + (Percentage/100 * Original Value)
A baker is using a European recipe that calls for an oven temperature of 180 degrees Celsius. The baker’s oven only displays temperatures in Fahrenheit. What temperature should the baker set the oven to?
f=9/5c+32 Identify the known value: c = 180
Substitute into the formula:
f = (9/5) * 180 + 32
Perform the multiplication:
f = (9 * 180) / 5 + 32
f = 1620 / 5 + 32
f = 324 + 32
Perform the addition:
f = 356
What is the formula to find the mean of a set of numbers?
Mean = (Sum of all values) / (Number of values)
