Question

solve.
1
one number is 6 more than the other. also, 7 times the smaller number is equal to 6 times the larger. find the two numbers.
2
lamera cut 48 tulips and 24 daffodils from her garden. if she wants to make identical flower arrangements with none left over, what is the greatest number of arrangements she can make? how many of each type of flower would be in each arrangement?

Explanation:

Problem 1

Step1: Define variables

Let the smaller number be \( x \), then the larger number is \( x + 6 \).

Step2: Set up equation

Given that \( 7x = 6(x + 6) \).

Step3: Solve the equation

Expand the right side: \( 7x = 6x + 36 \).
Subtract \( 6x \) from both sides: \( 7x - 6x = 6x + 36 - 6x \), so \( x = 36 \).

Step4: Find the larger number

The larger number is \( x + 6 = 36 + 6 = 42 \).

Step1: Find GCD of 48 and 24

The greatest common divisor (GCD) of 48 and 24. Since \( 48 = 24\times2 \), the GCD of 48 and 24 is 24.

Step2: Determine number of arrangements

The greatest number of arrangements is the GCD, so 24 arrangements.

Step3: Find number of each flower per arrangement

For tulips: \( \frac{48}{24} = 2 \) tulips per arrangement.
For daffodils: \( \frac{24}{24} = 1 \) daffodil per arrangement.

Answer:

The two numbers are 36 and 42.

Problem 2