To easily solve ratio problems and to gain a better understanding of the topic in general, it is convenient to know about equivalent ratios.
Equivalent ratios are, in fact, ratios that seem different, are not expressed in the same way but, by simplifying or expanding them, you arrive at exactly the same relationship.
José and Dani have notebooks and pencils. José has 4 notebooks and 8 pencils.
The ratio between the notebooks and pencils that Dani has is the same as José's. Dani has 6 notebooks. We are asked to calculate how many pencils Dani has.
We see that the number of pencils José has is double the number of notebooks he has. Since we already know that the ratio between notebooks and pencils that José and Dani have is identical, we can deduce that Dani has 12 pencils (6 times 2, so that the number of pencils is double the number of notebooks).
Examples and exercises with solutions of Equivalent Ratios
Exercise #1
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?
Video Solution
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
641
Exercise #2
The rectangle ABCD is shown below.
AB = X
The ratio between AB and BC is 2x.
The length of diagonal AC is labelled m.
Choose the correct answer.
Video Solution
Step-by-Step Solution
We know that:
BCAB=2x
We also know that AB equals X.
First, we will substitute the given data into the formula accordingly:
BCx=2x
x2=BCx
xx2=BC
xx×x×2=BC
x×2=BC
Now let's look at triangle ABC and use the Pythagorean theorem:
AB2+BC2=AC2
We substitute in our known values:
x2+(x×2)2=m2
x2+x×2=m2
Finally, we will add 1 to both sides:
x2+2x+1=m2+1
(x+1)2=m2+1
Answer
m2+1=(x+1)2
Exercise #3
Given the rectangle ABCD
AB=X the ratio between AB and BC is equal to2x
We mark the length of the diagonal A with m
Check the correct argument:
Video Solution
Step-by-Step Solution
Let's find side BC
Based on what we're given:
BCAB=BCx=2x
BCx=2x
2x=xBC
Let's divide by square root x:
x2×x=BC
x2×x×x=BC
Let's reduce the numerator and denominator by square root x:
2x=BC
We'll use the Pythagorean theorem to calculate the area of triangle ABC:
AB2+BC2=AC2
Let's substitute what we're given:
x2+(2x)2=m2
x2+2x=m2
Answer
x2+2x=m2
Exercise #4
There are 18 balls in a box, 32 of which are white.
How many white balls are there in the box?
Video Solution
Answer
12
Exercise #5
In a box there are 28 balls, 41 of which are orange.
How many orange balls are there in the box?
Video Solution
Answer
7
Do you know what the answer is?
Question 1
There are two circles.
The length of the diameter of circle 1 is 4 cm.
The length of the diameter of circle 2 is 10 cm.
How many times larger is the area of circle 2 than the area of circle 1?