# Translate into mathematical form

#### bergausstein

##### Active member
if the greater of two numbers is one more than twice the smaller and the smaller is x, what is the bigger? if the bigger is x, what is the smaller number?

my answer to the first part is 2x+1.

but i don't know how to represent the second condition in a mathematical form.

and can you also give me the meaning of this phrase "there are twice as many x as there are y." or at least give an equivalent interpretation of it.

thanks.

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#### MarkFL

Staff member
For the second condition, let the smaller number by $y$, and so we have:

$$\displaystyle 2y+1=x$$

Now solve for $y$ to find the smaller number $y$ in terms of the larger number $x$.

#### bergausstein

##### Active member
the smaller number is $\frac{x-1}{2}$.

"there are twice as many x as y" can you give an equivalent statement for this.

#### MarkFL

Staff member
the smaller number is $\frac{x-1}{2}$.

"there are twice as many x as y" can you give an equivalent statement for this.
How do you have to change $y$ so that the result is equal to $x$?

#### bergausstein

##### Active member
How do you have to change $y$ so that the result is equal to $x$?
2x=y? i this correct?

and oh, going back to the first question why did you change $x$ to $y$ in $2x+1$?

#### MarkFL

Staff member
2x=y? i this correct?

and oh, going back to the first question why did you change $x$ to $y$ in $2x+1$?
No, that's not correct. If there are twice as many $x$ as $y$, then we have to double $y$ to be equal to $x$.

I'm not sure what you mean...I didn't change $x$ to $y$. I let $y$ be the smaller number. We have to double that smaller number, and then add 1 and this is equal to the larger number, hence:

$$\displaystyle 2y+1=x$$

Solving this for $y$ will give you the smaller number in terms of the larger.

#### bergausstein

##### Active member
my understanding to this phrase"there are twice as many x as y" is for every one $y$ there's two $x$. that's why i think about $2x=y$ as the mathematical form of it.

#### MarkFL

Staff member
my understanding to this phrase"there are twice as many x as y" is for every one $y$ there's two $x$. that's why i think about $2x=y$ as the mathematical form of it.
Suppose you have twice as many apples as oranges. If $x$ is the number of apples and $y$ is the number of oranges, how many oranges do you need so that you have an equal number of each?

#### bergausstein

##### Active member
i'll double the number of oranges?

#### MarkFL

Staff member
i'll double the number of oranges?
Correct, you need to add the number of oranges already there so that the number of oranges and apples is the same:

$$\displaystyle x=y+y=2y$$

This means $x$ is twice is big as $y$, or "there are twice as many $x$ as there are $y$."

#### bergausstein

##### Active member
hmm. if i let $y=\,number \,oranges$ then $2y = number, \,of\, apples$.

or if i let $y = \,number \,of \,apples$ then $\frac{y}{2}=\,number \,oranges$

are these correct?

#### MarkFL

Staff member
hmm. if i let $y=\,number \,oranges$ then $2y = number, \,of\, apples$.

or if i let $y = \,number \,of \,apples$ then $\frac{y}{2}=\,number \,oranges$

are these correct?
Yes, that's correct.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
"there are twice as many x as there are y."
I was uncomfortable reading this phrase because x is used to denote two different things: a category of objects ("apples") and the number of objects of this category (number of apples). I would pose the problem as follows. "There twice as many A's as B's. Let x be the number of A's and y be the number of B's. How are x and y related?"

#### Jameson

Staff member
It's important to note that there are equations here to find and they are not together a system of equations. It's more like Case 1 and Case 2.

Case 1: $x<y$

$2x+1=y$

The bigger, $y$ is one more than twice the smaller, $x$.

Case 2: $y<x$

______________________________?

Setting up this kind of equation was always a challenge for the high school student's I tutored for standardized tests in the past. The best way I was able to relate this was to first set up something like this:

$\_ \cdot x + \_=\_ \cdot y+\_$

There are four possible blanks to try to even out the equation. If we know one side is bigger than the other then we have to make the smaller side bigger somehow.

#### bergausstein

##### Active member
It's important to note that there are equations here to find and they are not together a system of equations. It's more like Case 1 and Case 2.

Case 1: $x<y$

$2x+1=y$

The bigger, $y$ is one more than twice the smaller, $x$.

Case 2: $y<x$

______________________________?

Setting up this kind of equation was always a challenge for the high school student's I tutored for standardized tests in the past. The best way I was able to relate this was to first set up something like this:

$\_ \cdot x + \_=\_ \cdot y+\_$

There are four possible blanks to try to even out the equation. If we know one side is bigger than the other then we have to make the smaller side bigger somehow.
thanks for the reply jameson. but in my book i'm restricted to just use one variable. how will i do that without using another variable?

i kind of understand the first case. but i'm having a hard time with this "$\_ \cdot x + \_=\_ \cdot y+\_$ what do you mean?

#### Jameson

Staff member
thanks for the reply jameson. but in my book i'm restricted to just use one variable. how will i do that without using another variable?

i kind of understand the first case. but i'm having a hard time with this "$\_ \cdot x + \_=\_ \cdot y+\_$ what do you mean?
I realize it might be unclear where I'm going with that so I'll use addition as an example. Let's say we're given a statement - "John is 5 years older than Frank". How can we express this mathematically? I would start by choosing two letters to represent each of their ages and do the following:

$J+\_=F+\_$.

I know that the above is not true since they are not the same age so I need to even them out. I've tried to draw two blank spaces on both sides where 5 could possibly go. It will go in one of the blanks, but which one? Lots of people will add 5 to the wrong side here.

Which one is true?

1) $J+5=F$
2) $J=F+5$

This is an easy example but the idea is important and will help with lots of word problems where you need to form equations correctly before you can solve them.

Anyway, about your question. What equation do you get if $y<x$?

#### Jameson

Ok I think I see what you mean by using one variable. We can write everything in terms of $x$ for each case. I feel I have derailed this thread slightly so here's my solution for others to check.
Case 1: $x<y$
$2x+1=y$ so the smaller number is $x$ and the bigger number is $2x+1$.
Case 2: $x>y$
$2y+1=x$ so the smaller number is $\dfrac{x-1}{2}$ and the bigger number is $x$.