Susan had 117 beads and marbles altogether. The beads and marbles were either white or black. $20\%$ of the beads and $50\%$ of the marbles were white. There were as many black beads as black marbles.

1. Represent the beads and marbles in terms of their white and black percentages.

Let $B$ be the total number of beads and $M$ be the total number of marbles.
We are given:

• Beads: $20\%$ are white, so $100\% - 20\% = 80\%$ are black.
  $\text{Black beads} = 0.8B$
• Marbles: $50\%$ are white, so $100\% - 50\% = 50\%$ are black.
  $\text{Black marbles} = 0.5M$

2. Equate the number of black beads and black marbles to find the ratio of total beads to total marbles.

Since the number of black beads is equal to the number of black marbles:
$0.8B = 0.5M$
$\frac{B}{M} = \frac{0.5}{0.8} = \frac{5}{8}$

3. Compare the number of beads and marbles to choose the correct phrase.

Since the ratio of beads to marbles is $5 : 8$, the number of beads ($5$ units) is less than the number of marbles ($8$ units). Therefore, the number of beads Susan had was lower than the number of marbles.

1. Express the relationship between the white and black parts of beads and marbles using percentages and ratios.

We are given that:

• $20\%$ of the beads are white, which means $100\% - 20\% = 80\%$ of the beads are black.
• $50\%$ of the marbles are white, which means $100\% - 50\% = 50\%$ of the marbles are black.

Let the total number of beads be $B$ and the total number of marbles be $M$.
We are given that the number of black beads equals the number of black marbles:
$80\% \text{ of } B = 50\% \text{ of } M$
$\frac{80}{100} B = \frac{50}{100} M \implies 8B = 5M$

2. Find the ratio of the number of beads to marbles and allocate units to the total.

From $8B = 5M$, we get the ratio of beads to marbles:
$\frac{B}{M} = \frac{5}{8}$
So, the ratio of beads to marbles is $5 : 8$.

Let the number of beads be $5u$ and the number of marbles be $8u$.
Total number of beads and marbles:
$5u + 8u = 13u$
Given total is 117:
$13u = 117$

3. Solve for 1 unit and calculate the number of beads.

Find the value of $1u$:
$u = 117 \div 13 = 9$

Calculate the number of beads ($5u$):
$\text{Number of beads} = 5 \times 9 = 45$