c - j = 11
c = 2j - 16
2j - 16 - j = 11
j = 27
c = 2(27) - 16 = 38
Desiree G.
asked • 02/16/17Chris is older than jorge. The difference between their ages is 11years. Chris is 16years less than doubled jorges age. How old is chris and how old is jorge.
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Willa R. answered • 02/16/17
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This problem may seem an intimidating and convoluted mess, but we can break it down into two equations, making it easier to work with. Let C be the age for Chris and J be Jorge's age, both in years.
Taking the first two sentences, Chris is 11 years older than Jorge, meaning
C=11+J.
Now we have to deal with the fact that Chris is 16 years younger than the double of Jorge's age. The double of Jorge's age is 2*J, and we need 16 years less than that. Therefore, the second equation is as follows.
C=2*J-16
This is two equations for two unknowns. Remember, you always need as many equations as variables (unknowns). You must then use one equation to reduce the number of variables in the other equation by 1, allowing you to solve for one of the variables.
Solve the second equation for J. 2*J=C+16 => J=C/2+8
Now substitute into the first equation. C=11+C/2+8. This creates a single variable equation, which is readily solvable.
Put everything with C on one side. C-C/2=19
Simplify: C/2=19
Multiply both sides by 2: C=19*2=38
Recall that C=11+J. Solve for J. J=C-11=38-11=27.
Check: 27*2=54. 54-16=38.
Taking the first two sentences, Chris is 11 years older than Jorge, meaning
C=11+J.
Now we have to deal with the fact that Chris is 16 years younger than the double of Jorge's age. The double of Jorge's age is 2*J, and we need 16 years less than that. Therefore, the second equation is as follows.
C=2*J-16
This is two equations for two unknowns. Remember, you always need as many equations as variables (unknowns). You must then use one equation to reduce the number of variables in the other equation by 1, allowing you to solve for one of the variables.
Solve the second equation for J. 2*J=C+16 => J=C/2+8
Now substitute into the first equation. C=11+C/2+8. This creates a single variable equation, which is readily solvable.
Put everything with C on one side. C-C/2=19
Simplify: C/2=19
Multiply both sides by 2: C=19*2=38
Recall that C=11+J. Solve for J. J=C-11=38-11=27.
Check: 27*2=54. 54-16=38.
Willa R.
In algebra, we can alter equations and have them still work by simply doing the same thing on both sides or by switching out one expression for an equivalent one. The key insight to understanding this problem is that the two ages can be considered variables, and that the sentences can be translated into equations, going from words to mathematics. Mathematics is ultimately just another language.
From there it's a matter of changing the equations in ways that don't make the equation false, in order to make them more convenient to use, and separating the variables so that they don't share the equation with another variable (this is known solving the equations). When solving for two variables with two equations, there are several ways to do it, but the easiest to understand, and the one I used in the math above, is to rearrange one equation so that a variable is all by itself on the left side and then realize that you can use the new expression in place of the other, just like a synonym in a language.
Thus, when I had 2*J=C+16 (or, in English, Chris is 16 years younger than twice the age of Jorge), I wanted to either get J on one side or C on one side, with nothing else. I chose to do this by dividing both sides by 2. 2*J/2 is J*2/2, and 2/2=1, just as any number divided by itself is 1. But I can't do this to the left side and leave the right side as just C+16. That would be like giving one person an apple and another two apples and saying that they have the same number of apples. So, I divide C+16 by 2. (C+16)/2 = C/2+16/2 by the law of distribution (meaning you can split the division across the addition and it still works). 16/2 is 8, by simple arithmetic. This transforms the equation into J=C/2+8, which in English is "Jorge is 8 years older than half of Chris's age." Everything on the right side lacks J, and J is the only thing on the left side, making it easier to work with in the next step.
So now I look at the other part of the word problem. The problem states that Chris is older than Jorge by 11 years. I make an equation for that. C=11+J. But we know that J=C/2+8, so we can replace any J that we see with C/2+8 and not make the equation untrue, just like you can switch out a word with its synonym in English and the sentence means the same thing. So we do that. C=11+J becomes C=11+C/2+8, or C=19+C/2, or "Chris is 19 years older than half of his own age," to make it simpler.
Now we want to get that C by itself, like we did with J in the other equation. We can subtract C/2 from both sides of the equation without changing its meaning. C=19+C/2 becomes C-C/2=19+C/2-C/2. "Chris's age minus half of the same age is 19 years older than the age you get if you first put in half his age and then take it away again, leaving nothing." I will be the first to admit that's a clunky sentence. It's also a clunky equation. We cancel out the C/2's on the right side since they mean nothing. C-C/2=19. "If you take away half of Chris's age, he would be 19." Then we have to realize that a whole C is two halves of a C, just like anything is, so if you take a half away from a whole you get the other half. C/2=19. "Someone who is 19 would be half Chris's age." Now we double both sides, which in algebra is a valid step because you are doing the exact same thing on both sides. 2*C/2=2*19. "Double half of Chris's age is double 19." You can see that "double half" is another phrase that just undoes itself. The sentence could have just said "Chris's age is twice as much as 19" and meant the same thing. In mathematics, then, the 2's on the left cancel each other out for the same reason that "double of half" is a phrase that renders itself useless.
C=2*19=2*10+2*9=20+18=38, or you can simply use a calculator to get that part.
Lastly, we go right back to that first statement. Chris is 11 years older than Jorge. Someone who is 11 years younger than a 38 year old is 27.
Does that clear things up?
From there it's a matter of changing the equations in ways that don't make the equation false, in order to make them more convenient to use, and separating the variables so that they don't share the equation with another variable (this is known solving the equations). When solving for two variables with two equations, there are several ways to do it, but the easiest to understand, and the one I used in the math above, is to rearrange one equation so that a variable is all by itself on the left side and then realize that you can use the new expression in place of the other, just like a synonym in a language.
Thus, when I had 2*J=C+16 (or, in English, Chris is 16 years younger than twice the age of Jorge), I wanted to either get J on one side or C on one side, with nothing else. I chose to do this by dividing both sides by 2. 2*J/2 is J*2/2, and 2/2=1, just as any number divided by itself is 1. But I can't do this to the left side and leave the right side as just C+16. That would be like giving one person an apple and another two apples and saying that they have the same number of apples. So, I divide C+16 by 2. (C+16)/2 = C/2+16/2 by the law of distribution (meaning you can split the division across the addition and it still works). 16/2 is 8, by simple arithmetic. This transforms the equation into J=C/2+8, which in English is "Jorge is 8 years older than half of Chris's age." Everything on the right side lacks J, and J is the only thing on the left side, making it easier to work with in the next step.
So now I look at the other part of the word problem. The problem states that Chris is older than Jorge by 11 years. I make an equation for that. C=11+J. But we know that J=C/2+8, so we can replace any J that we see with C/2+8 and not make the equation untrue, just like you can switch out a word with its synonym in English and the sentence means the same thing. So we do that. C=11+J becomes C=11+C/2+8, or C=19+C/2, or "Chris is 19 years older than half of his own age," to make it simpler.
Now we want to get that C by itself, like we did with J in the other equation. We can subtract C/2 from both sides of the equation without changing its meaning. C=19+C/2 becomes C-C/2=19+C/2-C/2. "Chris's age minus half of the same age is 19 years older than the age you get if you first put in half his age and then take it away again, leaving nothing." I will be the first to admit that's a clunky sentence. It's also a clunky equation. We cancel out the C/2's on the right side since they mean nothing. C-C/2=19. "If you take away half of Chris's age, he would be 19." Then we have to realize that a whole C is two halves of a C, just like anything is, so if you take a half away from a whole you get the other half. C/2=19. "Someone who is 19 would be half Chris's age." Now we double both sides, which in algebra is a valid step because you are doing the exact same thing on both sides. 2*C/2=2*19. "Double half of Chris's age is double 19." You can see that "double half" is another phrase that just undoes itself. The sentence could have just said "Chris's age is twice as much as 19" and meant the same thing. In mathematics, then, the 2's on the left cancel each other out for the same reason that "double of half" is a phrase that renders itself useless.
C=2*19=2*10+2*9=20+18=38, or you can simply use a calculator to get that part.
Lastly, we go right back to that first statement. Chris is 11 years older than Jorge. Someone who is 11 years younger than a 38 year old is 27.
Does that clear things up?
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02/16/17
Kurt T. answered • 02/16/17
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(162)
Math Tutoring and Test Prep
C = Chris, J = Jorge.
C = J + 11
C = 2J - 16
C = J + 11
C = 2J - 16
Since both (J + 11) and (2J - 16) are equal to C, they must also be equal to each other.
J + 11 = 2J - 16
J = 27
Jorge is 27, Chris is 38.
Jorge is 27, Chris is 38.
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Desiree G.
02/16/17