A positive integer is 6 less than another. If the reciprocal of the smaller integer is subtracte

A positive integer is 6 less than another. If the reciprocal...

Key Concepts

Constraints on Integers

Identifying that the solution must satisfy specific conditions such as being positive integers is important. This concept ensures that only valid solutions are considered in the context of the problem.

Equation Solving

Once the problem is translated into algebraic equations, methods for solving linear or fractional equations are applied. This concept includes isolating variables, combining like terms, and manipulating fractions to find the unknown values.

Reciprocal Relationships

Understanding the concept of reciprocals, which are the inversion of numbers (i.e., 1 divided by the number), is crucial. This concept comes into play when the problem involves operations using the reciprocals of the quantities.

Word Problem Translation

This involves converting the descriptive language of a problem into algebraic expressions and equations. Recognizing relationships described in words and assigning variables to unknown quantities is a key step in formulating the problem mathematically.

Transcript

00:01 We're told that a positive integer is six less than another, and if the reciprocal of the smaller integer is subtracted from 10 times the reciprocal of the larger, then the result is 3 -7th.

00:12 And they want us to find the two integers.

00:14 So let's start by defining our variables.

00:17 So let's say that we have x and y.

00:21 And let's say that y is going to be the second one where it's six less than large ones.

00:27 So this is going to be the smaller of the two numbers.

00:30 And x will be the larger of the two numbers.

00:33 And so we're told that a positive integer is six less than another.

00:38 So if y is the smaller one, that means that y is equal to six less than x.

00:44 So we can write an equation that represents that.

00:46 And then we're told that the reciprocal of the smaller one, so the reciprocal of y being one over y, is when it's subtracted from 10 times the reciprocal of the larger, we get three sevens.

01:00 So the reciprocal of y would be one over y, and it's going to be subtracted from 10 times the reciprocal of the larger number, so 1 over x.

01:09 And when i do that, i get three sevenths.

01:12 So now i have two equations.

01:14 Basically, i have a system of equations that i can now solve for the two integers.

01:19 And so given that y equals x minus six, so this part right here, we're going to substitute this in for y into our equation.

01:27 And so what that's going to look like.

01:29 So we can simplify 10 times the reciprocal of one over x.

01:32 It's just going to be 10 over x.

01:34 And then we're going to have one over and we're going to put x minus six there where y used to be.

01:41 And that value is going to equal three sevenths.

01:44 And so what we can do now, if we think about this as fractions, we can make common denominators and we can solve this.

01:51 So the common denominator is going to be x minus six.

01:54 And so we're going to have 10 times the quantity x minus six.

01:56 And so we're going to have 10 times the quantity x minus six.

02:00 Over x times the quantity x minus six minus to make the second one the same we need to multiply by x so one times x is x and then x times the quantity x minus six so now those two fractions have common denominators and we know that the value one subtracted is equal to three sevens and so what we're going to do now is we're going to combine this top piece we're going to simplify this numerator here and so ten times x minus 6 is 10x minus 60 minus x all over x times x would be x squared and then x times negative 6 would be negative 6x.

02:42 We've now simplified the left side of the equation and that's going to equal three sevenths.

02:47 And so we have essentially a proportion.

02:50 So i'm going to cross multiply here.

02:51 So i'm going to do three times this value and seven times this value...

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