Can You Solve This Math Riddle Involving Positive Numbers?

In summary, HallsofIvy trainees find that multiplying by 90x(x-3) eliminates fractions in equations, making the solution easier to find.
  • #1
Paulo2014
81
0
The difference between two positive numbers is 3.
The difference between their reciprocals is 1/90
What are the two numbers?


I worked out that:
x-y=3 and
1/x-1/y= 1/90
Is that right so far?
 
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  • #2
yes, keep going
 
  • #3
I don't have a clue what to do next
 
  • #4
why don't you just make a guess?

When you have two equations of two unkowns, the strategy is to express one of the unkowns as a "function" of the other unknown.

e.g

y+2x=9 gives: y = 9 -2x

Then substitue that into the other equation.
 
  • #5
Paulo2014 said:
The difference between two positive numbers is 3.
The difference between their reciprocals is 1/90
What are the two numbers?


I worked out that:
x-y=3 and
1/x-1/y= 1/90
Is that right so far?
NO! If x- y= 3, a positive number, then x must be greater than y. But taking reciprocals reverses order. If x> y then 1/x< 1/y so 1/x- 1/y is negative. Assuming that x is the larger of the two numbers so that x- y= 3, then you must have 1/y- 1/x= 1/90.

Now you know that y= x- 3 so 1/(x- 3)- 1/x= 1/90. I would recommend multiplying that every term in that equation by 90x(x-3) in order to get rid of the fractions.

By the way, it is always a good idea to start a problem like this, NOT by saying "x- y= 3" but by defining the variables: "Let x and y be the two numbers, with [itex]x\ge y[/itex]".
 
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  • #6
why would I need to multiply everything by 90x(x-3)? I understand about the 1/y- 1/x= 1/90 part but I don't understand anything else...
 
  • #7
Paulo2014 said:
why would I need to multiply everything by 90x(x-3)? I understand about the 1/y- 1/x= 1/90 part but I don't understand anything else...

You multiply by 90x(x-3) in order to cancel the terms in the denominator, making it easier to solve for x.
 
  • #8
OK, so far you should understand that the 2 equations you are working with are:
a: x - y = 3
b: 1/y - 1/x = 1/90

What *I* would do if I'm a rookie is multiply equation b by x*y*90 since that is the LCM of the denominators and I'm scared of fractions. This would give me:
b*: 90x - 90y = xy

at this point you can use the substitution method to solve.

HallsofIvy had a slightly different approach which comes with training. First he solved eqn a for y. ie a*: y = x -3. Then he substituted this into equation b to get:
b*: 1/(x-3) - 1/x = 1/90

now the LCM of the denomiators is x * (x-3) * 90. If you mulitply by that you get:
b**: 90x - 90(x-3) = x(x-3)

If you follow the first method, you'll see that both methods turn out to be the same, except that my way is easier because I got rid of all fractions before doing any work. After some "training" though, you get used to the fractions, and just solve without getting rid of them the second you see them.

I think the biggest lesson to take away from this is be sure to label your variables at the beginning! So I'm double emphasizing it =)
 

1. What is a positive number math riddle?

A positive number math riddle is a puzzle that involves using positive numbers and basic mathematical operations (addition, subtraction, multiplication, division) to find a solution or answer. The challenge is to use critical thinking and problem-solving skills to come up with the correct answer.

2. How do you solve a positive number math riddle?

To solve a positive number math riddle, you need to first carefully read and understand the question. Then, you can use the given information and apply mathematical operations to find a solution. It may also help to write out the numbers and operations in a clear and organized way to avoid confusion.

3. Are there any tricks to solving positive number math riddles?

Yes, there are some common strategies or tricks that can help you solve positive number math riddles. These include looking for patterns, simplifying complex equations, and using estimation or rounding to get a ballpark answer. It's also helpful to think outside the box and consider multiple solutions.

4. Can positive number math riddles have more than one answer?

Yes, some positive number math riddles may have more than one possible solution. This often depends on the specific wording and details of the question. In some cases, there may be a "trick" answer that is different from the obvious solution. It's important to carefully analyze the question and consider all possibilities.

5. Why are positive number math riddles important?

Positive number math riddles are important because they encourage critical thinking, problem-solving skills, and creativity. They can also be a fun and engaging way to practice and reinforce mathematical concepts. Additionally, they can help improve mental math abilities and build confidence in approaching and solving complex problems.

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