Division in a given ratio - Examples, Exercises and Solutions

Understanding Division in a given ratio

Complete explanation with examples

What is division in a given ratio?

Division in a given ratio means splitting a total quantity into parts that maintain a specific proportional relationship, based on the ratio provided.
In a division according to a given ratio, we will have a defined quantity that we must divide according to said ratio. The process ensures that the ratio between the parts stays consistent, regardless of the total amount being divided. This concept is frequently used in various scenarios, such as dividing an inheritance, sharing resources, or solving problems in geometry.

Let's use an Example:

We want to divide 100100 Dollars in a 2:32:3 ratio.
So, the quantity is 100100 , and the ratio provided is 2:32:3 .

In order to do so, let's follow there simple steps:

  1. Add the parts of the ratio. In our case: 2+3=52 + 3 = 5 .
    Now we know that we need to divide the quantity to 55 .
  2. Divide the total amount by 55 . In our case: 100:5=20100:5=20
    So we get 2020 Dollars per part.
  3. Multiply each of the ratio side by the part.
    So: 202=4020\cdot2=40 , 320=603\cdot20=60 .

And so the 100100 Dollars is divided into 4040 Dollars and 6060 Dollars , maintaining the 2:32:3 ratio.

Diagram demonstrating division in a given ratio: total quantity of $100 divided in the ratio 2:3, resulting in $40 and $60. A simple visualization for understanding ratio-based division, featured in a math tutorial on dividing quantities in specific ratios.

Detailed explanation

Practice Division in a given ratio

Test your knowledge with 3 quizzes

How many times longer is the radius of the red circle than the radius of the blue circle?

210

Examples with solutions for Division in a given 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

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