the ratio of three number is 4 : 3 : 7 and sum of their square is 666. what is the value of the largest of the three numbers?
Answers
Answer:
it will be sum of ratio/666
14/666
47
per unit
largest ratio 7
so the answer is
47 x 7
=329
Step-by-step explanation:
Answer:
A good place to start is to setup some equations to describe the situation.
The first bit of info is that there are “three numbers”. We don't know what they are, so let's call them x, y, and z.
They're in a ratio 4:5:6. This means that one fourth of the first number (x) is the same size as one fifth of the second number (y), and also the same size as one sixth of the third number (z). So, with this we can setup some equations:
x/4=y/5=z/6
Or, written more traditionally, these three equations:
x/4=y/5
y/5=z/6
z/6=x/4
Pretty good start. There's one more bit of info we get. The sum of the largest and smallest number are equal to the sum of the other number and 55. First, which of x, y, and z will be the largest and smallest? We don't know the numbers yet, but we do know their ratio….
When numbers are compared by a ratio, we are comparing their sizes - how big or small is a number compared to another number. Ratios size numbers consistently, so of x, y, and z, where x is sized by 4, y is sized by 5, and z is sized by 6, which number will be the largest and which will be the smallest?
Let's look at some examples. Obviously the numbers 4, 5, 6 are in ratio 4:5:6…. What's the next sequence of integers in the ratio? You can get it by multiplying each number by a scaling factor! We'll start with the smallest integer factor after 1… 2.
8, 10, 12 is in the ratio 4, 5, 6. What's the next sequence? Multiply the ratio numbers by scaling factor 3 to get it - 12, 15, 18. In each case of our example, and always, because of the way multiplication works, the smallest number in the ratio will correspond to the smallest number in the end, and the largest corresponds to the largest.
So, X will be the smallest number and z will be the largest, and y is the “other” number (or the ”third” in the original question). Now we can setup another equation:
x+z=y+55
The largest plus the smallest equals the other plus 55.
Now we'll solve these equations by transforming the earlier equations and plugging them into the last equation. Y is on the right all by itself, so I'll try to turn x and z in that last equation into y's.
x/4=y/5
We want to get x by itself so we can replace x with whatever is left. Multiply both sides by 4.
x=4y/5
Cool, now we can plug 4y/5 in wherever we have x. Let's do it with z too.
z/6=y/5
z=6y/5
Awesome. Let's plug them into the “sums” equation.
x+z=y+55
4y/5+6y/5=y+55
Now we'll solve for y - we'll try to get y by itself and a number on the other side. First, the sum on the left has a common denominator, so we can combine numerators.
(4y+6y)/5=y+55
10y/5=y+55
2y=y+55
2y−y=55
y=55
So, we got y, which is the middle number! Now, we know that 55 corresponds to the 5 in the ratio. If we divide by 5, we can get the scaling factor for the other numbers in the ratio…. You can see this in the equations we setup, too. Let's get the scaling factor.
55/5=11
This is the number we can multiply by the other parts of ratio 4:5:6 to get all three numbers.
4∗11=44
5∗11=55
6∗11=66
So x, y, and z are 44, 55, 66, respectively, and your answer is 44, 55, 66.