Question

2 Math Questions! Well Medal!

1: http://gyazo.com/e7daee68cbde0f9de5f90acc299de09e

2: http://gyazo.com/96b5d3f9a3d843b71f728085327df659

Please explain how to do these questions, if you can. Thanks!

Answer 1

Rational numbers are contrasted with irrational numbers - such like Pi and square roots and sines and logarithms of numbers. This article concentrates on rational numbers, and at the end of the article you can click on a link to continue studying about irrational numbers.

In mathematical terms a number is rational if you can write it in a form a/b where a and b are integers, b not zero. Clearly all fractions are of that form. Terminating decimal numbers can easily be written in that form: for example 0.67 is 67/100, 3.40938 = 340938/100000 etc. You should review this with your child/students.

We can illustrate positive rational numbers with lines that go through the origin and another point with whole number coordinates. For example the line y = 2x has the slope 2 and it goes through the point (1,2). The line y = 3x goes through the point (1,3). The line y = 1/4x goes through the point (4,1). The line y = 2 1/2 x goes through the point (2,5). And, these points are the FIRST ones the lines go through after the origin.

Answer 2

For GMAT, we must know how to convert non-terminating repeating decimals into rational
numbers. We know how to do vice versa i.e. given a rational number, we can divide the numerator
by the denominator to find its decimal equivalent. For example,
1/3 = 0.333333333… (infinite number of 3s)
We write this as 0. .
Similarly, 1/6 = .16666666666… (infinite number of 6s)
We write this as 0.1 .
1/7 = 0.142857142857…
We write this as 0. .
The problem arises when we are given a decimal which we need to convert to a fraction. We know
that every non-terminating repeating decimal can be written cleanly and then used in calculations
by converting it to fraction. But given, say 1. , how do we know which fraction it represents? We
can approximate and say that 1.88888… is almost 1.9 but approximation may not be suitable in
every question or you might be asked for the actual value of the fraction.
So how do you convert 1. to a fraction? A terminating decimal is easy to handle. Say, a decimal
such as 1.8. We get rid of the decimal sign by dividing by 10 i.e. 1.8 = 18/10.
When we have a decimal such as 1. , the problem is that we have infinite 8s so we will need infinite
0s in the denominator.
1.888 = 1888/1000
1.888888 = 1888888/1000000
But what do we do when we have infinite 8s? It is very hard for us to fathom infinite numbers and
harder still to work with them. We need to get rid of the infinite sequence in some way. The good
thing about the infinite sequence is that even if we pull away one 8 out of it, the sequence still
remains infinite.
X = 1.
10X = 18. (When you multiply by 10, the decimal moves one place to the right but you still have
infinite 8s leftover)
Subtract the first equation from the second to get,
9X = 18. - 1. = 17 (Infinite 8s subtract out the infinite 8s and you are left with 18 – 1 = 17)
So X = 17/9
So the exact value of 1. is 17/9.

Answer 3

Question marks are ... (dot dot dot)

Answer 4

Good luck! :)

Answer 5

alright i get it now thanks.