Jeremy is playing a game called “Rational Round Up” where he has to collect all the numbers in a maze that are rational and then get to the end of the maze. When he collects a number, he must prove it is rational by writing it as a quotient of two integers. Help him determine how to prove that each of the following numbers is rational.
1. 2.4 = 12/5
2. 74 = 74/1 rational
3. 17.3333333… = 52/3 rational
4. π pi cant be expressed as a fraction so irrational.
5. 61/11 rational
6. –18 = -18/1 rational
7. = 6/10 rational
8. 87.125
9. –30 = -30/1 rational
10. –8.3
11. 58.25 = 58 ¼ rational
12. 121
13. 4.5
14. = 71/10 rational
I have completed a few i just need help on the rest.
8.
87.125 = 87125/1000 rational :)
same for 83/10
same for 121 / 1 :)
and for 45 / 10
Thanks! but could you also explain the process.
I just said if you could show it as a ratio of two whole numbers it is rational.
If it is a decimal fraction, that does it unless the number of digits required is infinite.
like
1.00001 = 100001 * 10^-5 = 100,001/ 100,000
now if it is pi or e or something, you can never settle on how many digits or what they are. (if they repeat like 3.13131313 ..... that can be done
but if they are unpredictable like
3.14159 .... who knows what
then they can not be expressed exactly by a ratio of whole numbers
of course an approximation like 22/7 is possible, but not exact.
Now maybe you are not there yet but for something like 3.13131313 ....
that is
3 + 13*10^-2 + 13*10^-4 + 13 * 10^-6
= 3 + geometric series a + ar + a r^2 + ar^3 ...
where a = 13*10^-2 and r = 10^-2
the sum of that series is a /(1-r)
13*10^-2 (1/(1-10^-2) = 13*10^-2/ 0.99
so 3 + 13/99 = (297+ 13)/99 = 310/99 =
which believe it or not is 3.13131313 ....
so if you can write it as 310/99
it is rational
but if you simply can not write it as a ratio of whole numbers, it is not rational.
Sure, I can help you! Let's go through each number and explain how to prove whether it is rational or not.
1. To prove that 2.4 is rational, we need to express it as a quotient of two integers. We can think of 2.4 as 2 and 4 tenths. Since 4 tenths is equivalent to 4/10, we can simplify it by reducing the fraction to lowest terms: 4/10 = 2/5. Therefore, 2.4 is rational and can be expressed as 12/5.
2. The number 74 is already in the form of an integer. Any integer can be expressed as a quotient of itself and 1. Therefore, 74 is rational.
3. The number 17.3333333... is a recurring decimal, with the digit 3 repeating infinitely. To express it as a rational number, we can use the pattern of the repeating digits. Since the decimal part repeats every 3 digits, we can multiply both sides of the equation by 10^3 (1000) to get rid of the repeating part. This gives us 1000 * 17.3333333... = 17333.3333333... and then subtracting the original equation (17.3333333...) from this new equation, we have 1000 * 17.3333333... - 17.3333333... = 17333.3333333... - 17.3333333..., which simplifies to 999 * 17.3333333... = 17316. Therefore, 17.3333333... can be expressed as 17316/999, making it rational.
4. The number π (pi) is an irrational number. It cannot be expressed as a quotient of two integers. Therefore, it is not rational.
5. The fraction 61/11 is already in the form of a quotient of two integers. Therefore, it is rational.
6. Similar to example 2, -18 is already an integer. Hence, it can be expressed as -18/1, making it rational.
7. The expression "= 6/10" is already written as a quotient of two integers. It is rational.
8. To determine whether 87.125 is rational, we need to express it as a quotient of two integers. We can write it as 87125/1000 by moving the decimal point three places to the right. Therefore, 87.125 is rational.
9. Similar to example 6, -30 is already an integer. It can be expressed as -30/1, making it rational.
10. To determine whether -8.3 is rational, we can express it as -83/10. By moving the decimal point one place to the right, we get -8.3 = -83/10. Therefore, -8.3 is rational.
11. The decimal 58.25 can be expressed as a mixed fraction. We can think of it as 58 whole units and 1/4. Therefore, 58.25 is rational and can be expressed as 58 1/4.
12. The number 121 is an integer and can be expressed as a quotient of itself and 1. Hence, 121 is rational.
13. The decimal 4.5 can be expressed as 9/2 by writing it as a fraction. Therefore, 4.5 is rational.
14. The expression "= 71/10" is already written as a quotient of two integers. Therefore, it is rational.
I hope this explanation helps you understand how to determine whether a number is rational or not. Let me know if you have any further questions!