(Note: I also included algebraic solution, where the missing values are represented by letters or variables so you can compare which is a faster and less error solution.)
Problem A:
I am thinking of two numbers. When I add the numbers, the answer is 10. When I subtract them, the difference is 2. What are the numbers:
Guess and Check Solution:
Possible Numbers:
a) 5 and 5
5 + 5 = 10 √
5 - 5 = 2 X
b) 8 + 2 = 10 √
8 - 2 = 2 X
c) 6 + 4 = 10 √
6 - 4 = 2 √
The numbers are 6 (the minuend) and 4 (the subtrahend).
Algebraic solution:
First addend: x (also the minuend)
Second addend: y (also the subtrahend)
Equation A: x + y = 10 Equation B: x - y = 2
Solution:
x + y = 10
y = 10 - x (use 10-x as subtrahend, instead of y)
Substitute 10 -x to y in Equation B:
x - (10-x) = 2
x + (-10) + x = 2 (Rule in subtraction of integers: Change the sign
each term in the subtrahend, then proceed to
addition.)
x + x - 10 = 2
2x = 2 + 10
2x/2 = 12
x = 6
First addend: x = 6
Second added: 10-x ⇒ 10-6 = 4
The numbers are 6 and 4.
Problem B:
Maine gave Lola Nidora a bouquet of 24 flowers. The bouquet was made up of roses, carnations, and daisies. There were twice as many carnations as roses, and three times as many daisies as roses. How many each kind of flower were in the bouquet?
Guess and check solution:
Roses: 1
Carnations: 2 (1) ⇒ twice as many as roses
Daisies: 3 (1) ⇒ thrice as many as roses
The ratio is 1 : 2 : 3
Guesses:
1 + 2 + 3 = 24 X
2 + 4 + 6 = 24 X
3 + 6 + 9 = 24 X
4 + 8 + 12 = 24 √
Answer: There were 4 roses, 8 carnations, and 12 daisies.
Algebraic solution:
Roses: x
Carnations: 2x
Daisies: 3x
Total Flowers: 24
Solution:
x + 2x + 3x = 24
6x = 24
6x/6 = 24/6
x = 4
The number of flowers for each kind:
Roses: x = 4 roses
Carnations: 2x = 2 (4) = 8 carnations
Daisies: 3x = 3 (4) = 12 daisies
There were 4 roses, 8 carnations, and 12 daisies.
