Two different positive whole numbers, a and b, are both greater than 1. The second number, b, is larger than the first number. After the second number is increased by 1, it is multiplied by the first number. The resulting product is 260. This information can be represented by the equation a x (b + 1) = 260.
Determine all possible pairs of numbers that satisfy the equation.
Hello,
ReplyDeleteSo I found 1 pair of numbers that satisfy the equation. 2x(129+1)=260. I got the numbers 2 and 129 by using reverse operations. Instead of adding and multiplying I used dividing and subtracting.
STEP 1: I know that 260 is a multiple of 2 and can be divided into 2 equally. So I divided 260 by 2. The variable "a" becomes 2.
STEP 2: Next I took the answer to 260/2 which was 130 and subtracted that by 1.130-1=129 therefore the variable "b" becomes 129.
STEP 3: To check my work, I plugged the numbers into the equation.
2x(129+1)=260
=129+1=130(ACCORDING TO BEDMAS, YOU HAVE TO DO BRACKETS FIRST.)
=130x2=260
Bye(to be continued with more pairs)
Hi,
ReplyDeleteOne possible pair of numbers I have found is
a=13
b=19
In order to get to this possible pair I used 2 different strategies, one is Order of Operations and the other one is Distributive Property.
ORDER OF OPERATIONS:
a x (b+1) = 260 For this process I have used BEDMAS (brackets, exponents, division,
= 13 x (19+1) multiplication, addition and subtraction). I used my answers and plugged them in
= 13 X 20 replacing (a) and (b). This strategy is used more often to find the answer to
= 260 equations.
DISTRIBUTIVE PROPERTY / ISOLATING :
ab + (a) = 260
a(b+1)=260 / b+1
260 / (b+1)
a = 260 / (b+1)
a = 260 / (19+1)
a = 260 / 20
a = 13
This strategy is more used to isolate or to find the number representing a variable(s). First since I know that a = 260 / (b+1). I took out the 0 out of 260 which gave me 26. Since I have to find the numbers for 2 variables I divided 26 by 2 to get 13. This tells me one of my variables is 13. Now that I have one of my variables, I can take that and divide it by 260 which gives me 20. Since I know I have to increase (b) by 1, I can work backwards and subtract 20 by 1 to get the number representing (b). Which is 19 = (20-1).
This pair meets the requirements:
- (b) is also a greater number than (a).
- Both (a) and (b) are positive whole numbers, larger than 1
So (a) is 13 and (b) is 19.
Thank you for reading my answer.Bye!
- Kajana Yokarajah
These are my strategies I have used I will add/continue on with more pairs using these strategies!
DeleteThank you bye!
-Kajana
Hello there!
ReplyDeleteThis is how I determined to solve the problem.
First, I listed common factors of 260. They are 1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130 and 260.
Next, I decided to take out 1 and 260 because in order to get 260 from both these numbers, you have to have the number 1 in the equation.
Then, I found the middle number in here: 2, 4, 5, 10, 13, 20, 26, 52, 65, 130
It was 13 and 20. So, starting from 20, I subtracted 1 from each number.
My new numbers, were 2, 4, 5, 10, 13, 19, 25, 51, 64, and 129.
Now, using these numbers, I can find all the possible pairs.
They are:
• 2 x (129 + 1) = 260
• 4 x (64 + 1) = 260
• 5 x (51 + 1) = 260
• 10 x (25 + 1) = 260
• 13 x (19 + 1) = 260
Hello. My brother and I got 5 different pairs.
ReplyDelete2 x (129 + 1) = 260 A = 2, B = 129
4 x (64 + 1) = 260 A = 4, B = 64
5 x (51 + 1) = 260 A = 5, B = 51
10 x (25 + 1) = 260 A = 10, B = 25
13 x (19 + 1) = 260 A = 13, B = 19
We used a combination of trial and error and common mathematical strategies to find these pairs.
We divided 260 by different numbers (based on our background knowledge ex. 2 because 260 is an even number and 260's factors) then subtracted one from the quotient to get the value of B. Then, A would be the divisor.
e.g
260 / 2 = 130
130 - 1 = 129
Therefore, 2 (divisor) is A, 130 (quotient) - 1 is B.
2 x (130 - 1) = 260
~ Rachel and Jonathan
Wow!
Deletei got the same thing! :)
Glad to see all these participations and seeing you all using terms like factors, multiples, and distributive property! =)
ReplyDeleteThank you!!! :)
DeleteIn considering only the equation a x (b + 1) = 260, we see that we are looking for two whole
ReplyDeletenumbers, each greater than 1, that multiply to 260. The first number is a and the second
number is a number that is 1 greater than b. We also want the second number to be greater
than the first number.
We can generate the following list of valid pairs of numbers that multiply to 260:
260 = 2 x 130 = 4 x 65 = 5 x 52 = 10 x 26 = 13 x 20:
There are other pairs of positive numbers that multiply to 260 but each of them can be
excluded because of the given restrictions.
We can exclude 260 = 1 x 260 and 260 = 260 x 1 because both numbers must be greater than 1.
We can also exclude 260 = 130 x 2, 260 = 65 x 4, 260 = 52 x 5, 260 = 26 x 10, and
260 = 20 x 13 because the second number in the product must be larger than the first number.
Since b is 1 less than the second number in each valid product, we can generate the pairs of
whole numbers that satisfy the equation a (b + 1) = 260. We will write the number pairs as
ordered pairs in the form (a; b).
The first ordered pair is (2; 129) since 2 x (129 + 1) = 2 x 130 = 260.
The second ordered pair is (4; 64) since 4 x (64 + 1) = 4 x 65 = 260.
The third ordered pair is (5; 51) since 5 x (51 + 1) = 5 x 52 = 260.
The fourth ordered pair is (10; 25) since 10 x (25 + 1) = 10 x 26 = 260.
The final ordered pair is (13; 19) since 13 x (19 + 1) = 13 x 20 = 260.
The five ordered pairs (a; b) that satisfy the equation a x (b + 1) = 260 as well as all the other
conditions are (2; 129), (4; 64), (5; 51), (10; 25), and (13; 19).