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I am currently read a book, and there is something that I'm struggling to understand (I have weak background in math):

Anya has 3 pounds of tofu, Betty has twice as much tofu as Anya, Carmen has 5 pounds more tofu than Betty, Deirdre has three times the tofu that Carmen has. In arithmetical it stands like that:

A= 3
B= 2XA
C= B+5
D= 3XC

Now for my problem of understanding, he writes:

We can combine these four statements into one statement by substitution and then finally perform the additions and multiplications:

D= 3XC
D= 3X(B+5)
D= 3X((2Xa)+5)
D= 3X((2X3)+5)
D= 33

My questions are:

  1. Why is the D stand for all of the letters before.
  2. What is the substitution he talks about.
  3. How he gets the number 33 at the end ?

I would like to get some background basic information about it.

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5 Answers 5

up vote 3 down vote accepted

He starts with the forth equation $$\tag{1}D=3 {C}$$ Then he takes the third equation $$\tag {2}C=B+5$$ and "substitutes" this into (1): equation (2) says that $C$ is the same quantity as $B+5$; so, in (1) we can replace the "$ {C}$" with $" {B}+5"$. This gives $$ \tag{3}D=3(B+5). $$ He then uses the equation $B=2A$. He takes the "$B$" in (3) and replaces it with "$2A$".

And so on.

At each stage, he always has $D$ on the left hand side.

At the end, he has the equation $$D=3\cdot((2\cdot3))+5.$$ The right hand side is just a number; doing the arithmetic gives $$ D=3\cdot((2\cdot3) +5)=3\cdot(6+5)=1\cdot11=33. $$

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as you wrote above we have $a=3,b=2a, c=b+5, d=3c$. the substitution you copied starts by taking the expression for $d$ and replacing the $c$ that appears by $b+5$ from the previous equation to get $d=3c=3(b+5)$. Next replace the $b$ that appears in this new expression for $d$ by $b=2a$ to get $d=3(b+5)=3(2a+5)$. Finally replace the $a$ with the number 3 to get $d=3(2a+5)=3(2\cdot3+5)=33$.

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$D$ should obviously represent the amount of tofu Deirrdre has (in pounds). All letters before occur in $D$, because the amount of tofu Deirrdre has indirectly depends on all other variables (the amount of tofu the other persons have).

By substitution he means using the prior equations to evaluate your equation for $D$. Namely you have $D=3C$, this means Deirdre has three times the tofu Carmen has, but we know that Carmen has 5 pound more tofu than Betty ($C=B+5$). Therefore we can substitute this into our equation for $D$, namely

$$D=3\cdot C=3\cdot (C)=3\cdot(B+5)$$

This is what he calls substitution. Iteratively using this method (inserting the equation for $B$ here and so on) you finally get

$$D= 3\cdot((2\cdot3)+5)$$

This term only contains numbers so you can simply evaluate it (note to evaluate brackets first). Then you will see that the result is 33.

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Oh well, so many answers at the same time :) –  Listing Dec 19 '11 at 21:49

D denotes the amount of tofu Deirdre has. You want to compute this amount and from the text you know that Dhas three times the tofu Carmen (C) has. They denote this by writing D= 3C.

Now you know that Chas $5$ pounds more tofu than B, that is, C= B + 5. You substitute it into C to get D = 3(B + 5).

You carry on in this manner until you have substituted all the information from the text, line by line, into your original equation D=3C and you end up getting 33, that is, you now know that Deirdre has 33 pounds of tofu.

To address 1.: The write D at the beginning of each line because each line denotes the amount of tofu has.

To address 2.: They are talking about substituting the information from the text into the equation for D.

Hope this helps.

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Anya has 3 pounds of tofu, Betty has twice as much tofu as Anya, Carmen has 5 pounds more tofu than Betty, Deirdre has three times the tofu that Carmen has.

Let's call $A$ for Anya, $B$ for Betty, $C$ for Carmen, and $D$ for Deirdre, so:

$A=3$ pounds of tofu

$B=2A = 2 \cdot 3= 6$ pounds of tofu

$C=B+5= 6+5= 11$ pounds of tofu

$D=3 \cdot C = 3 \cdot 11=33$ pounds of tofu

The substitution means put the letter and substitute it for its value to get one of the values, we can place first $A$ in $B$, then $B$ in $C$ and finally $C$ in $D$, like I did. Or we can try to replace from some other way like the author did, we take $D$ and we see that a $C$ is part of its calculation so we replace the $C$, for the value of $C$:

$D=3 \cdot C = 3(B+5)$. We get a $B$, so we can substitute its value for the one of $B$: $3(B+5)=3((2 \cdot A)+5)$, and now we replace $A$, for its value (3) to get

$D =3((2 \cdot A)+5)= 3((2(3))+5) = 33$ pounds, that is the total of pounds that Deidre has.

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