a math question?

Question

can someone please help me with this question?
the sum of two positive integers is eleven. what is the smallest possible sum of their reciprocals? i don't really get what they mean by reciprocals, so please help me
thanks a lot

Answer 1

Since they are both positive, the numbers are going to be one of the following pairs:
1, 10
2, 9
3, 8
4, 7
5, 6

The reciprocals are 1 over each of these numbers:
1/1 + 1/10
1/2 + 1/9
1/3 + 1/8
1/4 + 1/7
1/5 + 1/6

If you calculate each of these out, the smallest sum is 1/5 + 1/6

1/1 + 1/10 = 1.1
1/2 + 1/9 = 0.6111...
1/3 + 1/8 = 0.45833...
1/4 + 1/7 = 0.39285714...
1/5 + 1/6 = 0.3666...

So the numbers are 5 and 6.

Answer 2

Reciprocals are what happens when you turn a number upside down when it is in fraction form. So, since 5 as a fraction is 5/1. . .the reciprocal is 1/5! You can always check that you have the correct reciprocal because a number times its reciprocal will always equal 1. (5/1 * 1/5 = 1)

Now, for your problem you need to break down the question to figure out exactly what it is asking.

Sum--means you are adding something.

Two positive integers--whole numbers greater than zero, such as 1,2,3,4...

eleven--the two numbers have to add to eleven.

Now figure out what combinations of numbers can add to eleven: 1+10, 2+9, 3+8, 4+7, 5+6

But you need to find what happens when you add the reciprocals. So now change them to reciprocals and add them up. The smallest one will be the answer! (A good scientific calculator that does fractions could help here)

1/1+1/10=1.1
1/2+1/9=.611
1/3+1/8=.4583
1/4+1/7=.39285
1/5+1/6=.3666

Of course, on problems such as this you can save time by looking for patterns. If you noticed that the numbers were getting smaller as you tried them, you could have just jumped to the last one and saved a little time. I assume this is an SAT (or similar) question and even a few seconds extra time is always helpful!

The correct answer is .366 or .367 (depending on rounding).

Answer 3

A reciprocal is just 1 divided by the number, so the lowest sum would need the most in the denominator. For instance
1/1 +1/10 = 1-1/10, while 1/2 + 1/9 = 11/18, a smaller number. Keep looking

Answer 4

the sum of two positive integers is eleven. what is the smallest possible sum of their reciprocals?

let the numbers be a and b

a+b =11 from here b = 11 - a

a and b are positive

1/a +1/b =sum

our objective is to find a and b such that the sum is the smallest

sum = 1/a +1/(11-a)
= {(11-a) +a} /{a(11-a) =
{11}/{a(11-a)}

minimizing sum is equivalent to maximizing a(11-a)

derivative is = 11 -2a =0
when a =5.5

but we are limited to integers only
so a can only be = 5 or 6

let a =5 Sum = 11/{30} = 0.367
let a =6 Sum = 11/{30} = 0.367
the 2 integer positive numbers that minimize the reciprocals are
a =5
b =6

Answer 5

the sum of two positive integers is eleven. :
x+y=11

what is the smallest possible sum of their reciprocals?
Reciprocals is the inverse of the number for instance the inverse of 2 is 1/2...the inverse of 1/4 is 4...2/3--> 3/2
You *flip* the number basically....

so 1/x+1/y= smallest possible
Now x + y=11 has a variety of solutions...you want one that gives you the two biggest numbers (and therefore the two smallest fractions!!)

So 5 and 6 (doesn't matter which is x and y)

1/5+1/6= 11/30 or .36

Try 4 and 7 just to see if it gets bigger or smaller:

1/4+1/7= 11/28 or .39

It increases so you can conclude that 6 and 5 is right!

Answer 6

a reciprocal is the total opposite of the original number. For example: the reciprocal of 3 is -1/3, because you change the positive to negative and if you flip 3 (same as 3/1) you get 1/3.

Answer 7

for each answer x = 10 and y = 1 .....................(1)
= 9 and y = 2 ......................(2)
= 8 and y = 3 ......................(3)
........= 1 and y = 10 ......................(10)
so a reciprocal is 1/the number so for example
the reciprocal of 2 is 1/2 or one half or 0.5
and the reciprocal of 8 is 1/8 or one eighth or 0.125

if you check the sum of the reciprocals of the answers (1) to (10) you will be able to see which sum is the smallest

I have not included zero in my list of answers as division by zero is not defined in arithmetic.