Ratio Practice Problems - Master Ratios and Proportions

Practice ratio problems with step-by-step solutions. Learn to solve ratio word problems, convert ratios to fractions, and find missing values in proportions.

📚Master Ratio Problems with Interactive Practice
  • Read and interpret ratios correctly from left to right
  • Convert ratios between colon notation and fraction form
  • Solve ratio word problems involving two or more quantities
  • Find unknown values using ratio relationships and variables
  • Calculate part-to-whole ratios in real-world contexts
  • Verify ratio solutions by checking total quantities

Understanding Ratio

Complete explanation with examples

What is ratio?

The ratio describes the "relationship" between two or more things.

The ratio connects the given terms and describes how many times greater or smaller a certain magnitude is than another.

Let's see an example from everyday life:

When asked in a class, what is the ratio between boys and girls, it refers to how many girls there are in relation to a certain number of boys.

Or, for example, if in a certain vase there are red and white balls, the ratio between them can describe how many red balls there are in relation to a certain number of white balls or vice versa.

A - Ratio

Detailed explanation

Practice Ratio

Test your knowledge with 3 quizzes

How many times longer is the radius of the red circle (14 cm) than the radius of the blue circle, which has a diameter of 7?

Examples with solutions for Ratio

Step-by-step solutions included
Exercise #1

In a box there are 28 balls, 14 \frac{1}{4} of which are orange.

How many orange balls are there in the box in total?

Step-by-Step Solution

To solve this problem, we'll determine the number of orange balls by calculating the fraction of the total number of balls:

  • Step 1: Identify the total number of balls, 28 28 .
  • Step 2: Note the fraction representing the orange balls, 14 \frac{1}{4} .
  • Step 3: Apply the formula to find the number of orange balls:
    Number of orange balls =28×14 = 28 \times \frac{1}{4}

Now, let's perform the calculation:
28×14=28÷4=7 28 \times \frac{1}{4} = 28 \div 4 = 7

Therefore, the number of orange balls in the box is 7 7 .

Answer:

7

Video Solution
Exercise #2

If there are 18 balls in a box of which 23 \frac{2}{3} are white:

How many white balls are there in the box in total?

Step-by-Step Solution

To solve this problem, we will determine the number of white balls in the box using a fraction of the total number of balls.

We are given the total number of balls in the box as 18, and we know that 23 \frac{2}{3} of these balls are white. To find the number of white balls, we follow these steps:

  • Step 1: Identify the total quantity, which is 18 balls.
  • Step 2: Use the given fraction 23 \frac{2}{3} to find the number of white balls.
  • Step 3: Multiply the total number of balls by the fraction of white balls: 18×23 18 \times \frac{2}{3} .

Perform the calculation:

18×23=18×0.6667=12 18 \times \frac{2}{3} = 18 \times 0.6667 = 12

Alternatively, calculate directly using fractions:

18×23=18×23=363=12 18 \times \frac{2}{3} = \frac{18 \times 2}{3} = \frac{36}{3} = 12

Thus, the total number of white balls in the box is 12.

Therefore, the correct answer is choice 12.

Answer:

12

Video Solution
Exercise #3

There are two circles.

One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.

How many times greater is the area of the second circle than the area of the first circle?

Step-by-Step Solution

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

π*4² =

π16

Circle 2:

π*10² =

π100

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

100/16 =

6.25

Therefore the answer is 6 and a quarter!

Answer:

614 6\frac{1}{4}

Video Solution
Exercise #4

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal tox2 \sqrt{\frac{x}{2}}

We mark the length of the diagonal A A with m m

Check the correct argument:

XXXmmmAAABBBCCCDDD

Step-by-Step Solution

Let's find side BC

Based on what we're given:

ABBC=xBC=x2 \frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

2x=xBC \sqrt{2}x=\sqrt{x}BC

Let's divide by square root x:

2×xx=BC \frac{\sqrt{2}\times x}{\sqrt{x}}=BC

2×x×xx=BC \frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC

Let's reduce the numerator and denominator by square root x:

2x=BC \sqrt{2}\sqrt{x}=BC

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Let's substitute what we're given:

x2+(2x)2=m2 x^2+(\sqrt{2}\sqrt{x})^2=m^2

x2+2x=m2 x^2+2x=m^2

Answer:

x2+2x=m2 x^2+2x=m^2

Video Solution
Exercise #5

The rectangle ABCD is shown below.

AB = X

The ratio between AB and BC is x2 \sqrt{\frac{x}{2}} .


The length of diagonal AC is labelled m.

XXXmmmAAABBBCCCDDD

Determine the value of m:

Step-by-Step Solution

We know that:

ABBC=x2 \frac{AB}{BC}=\sqrt{\frac{x}{2}}

We also know that AB equals X.

First, we will substitute the given data into the formula accordingly:

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

x2=BCx x\sqrt{2}=BC\sqrt{x}

x2x=BC \frac{x\sqrt{2}}{\sqrt{x}}=BC

x×x×2x=BC \frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC

x×2=BC \sqrt{x}\times\sqrt{2}=BC

Now let's look at triangle ABC and use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

We substitute in our known values:

x2+(x×2)2=m2 x^2+(\sqrt{x}\times\sqrt{2})^2=m^2

x2+x×2=m2 x^2+x\times2=m^2

Finally, we will add 1 to both sides:

x2+2x+1=m2+1 x^2+2x+1=m^2+1

(x+1)2=m2+1 (x+1)^2=m^2+1

Answer:

m2+1=(x+1)2 m^2+1=(x+1)^2

Video Solution

Frequently Asked Questions

How do you read a ratio correctly?

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Read ratios from left to right, just like English. In the ratio 3:2, the first number (3) corresponds to the first item mentioned, and the second number (2) corresponds to the second item mentioned.

What's the difference between ratio notation and fraction notation?

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Ratios can be written as 3:2 (colon notation) or 3/2 (fraction notation). Both express the same relationship - for every 3 of the first item, there are 2 of the second item.

How do you solve ratio word problems with totals?

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Follow these steps: 1) Identify what the ratio represents, 2) Use a variable (like X) for each part, 3) Set up an equation adding all parts to equal the total, 4) Solve for X, 5) Calculate actual quantities by multiplying X by each ratio number.

Can ratios have more than two numbers?

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Yes, ratios can compare three or more quantities. For example, if red:blue:green balls are in a 2:3:1 ratio, it means for every 2 red balls, there are 3 blue balls and 1 green ball.

What does it mean when a ratio is 2:1?

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A 2:1 ratio means there are twice as many of the first item compared to the second item. For example, if pens to markers is 2:1, there are 2 pens for every 1 marker.

How do you check if your ratio answer is correct?

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Verify by adding up all the quantities to see if they equal the given total. Also check that the individual quantities maintain the original ratio relationship when simplified.

What are some common ratio mistakes students make?

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Common mistakes include: mixing up the order of terms when reading ratios, forgetting to use variables when solving for unknowns, and not checking that final answers add up to the given total.

When do we use ratios in real life?

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Ratios appear in cooking recipes, mixing paint colors, comparing boys to girls in a class, describing geometric shapes, financial investments, and sports statistics like win-loss records.

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