To determine if the given pairs of ratios are proportional, we'll go through each pair and check the relationship between them.
### Pair 1: [tex]$5: 15$[/tex] and [tex]$10: 30$[/tex]
A pair of ratios [tex]\( \frac{a}{b} \)[/tex] and [tex]\( \frac{c}{d} \)[/tex] are proportional if [tex]\( a \times d = b \times c \)[/tex].
For [tex]\( 5: 15 \)[/tex] and [tex]\( 10: 30 \)[/tex]:
[tex]\[ 5 \times 30 = 150 \][/tex]
[tex]\[ 15 \times 10 = 150 \][/tex]
Since both products are equal, the ratios are proportional.
Answer: yes
### Pair 2: [tex]$16: 4$[/tex] and [tex]$25: 5$[/tex]
For [tex]\( 16: 4 \)[/tex] and [tex]\( 25: 5 \)[/tex]:
[tex]\[ 16 \times 5 = 80 \][/tex]
[tex]\[ 4 \times 25 = 100 \][/tex]
Since the products are not equal, the ratios are not proportional.
Answer: no
### Pair 3: [tex]$10: 15$[/tex] and [tex]$20: 25$[/tex]
For [tex]\( 10: 15 \)[/tex] and [tex]\( 20: 25 \)[/tex]:
[tex]\[ 10 \times 25 = 250 \][/tex]
[tex]\[ 15 \times 20 = 300 \][/tex]
Since the products are not equal, the ratios are not proportional.
Answer: no
### Pair 4: [tex]$20: 60$[/tex] and [tex]$30: 90$[/tex]
For [tex]\( 20: 60 \)[/tex] and [tex]\( 30: 90 \)[/tex]:
[tex]\[ 20 \times 90 = 1800 \][/tex]
[tex]\[ 60 \times 30 = 1800 \][/tex]
Since both products are equal, the ratios are proportional.
Answer: yes
### Pair 5: [tex]$5: 10$[/tex] and [tex]$100: 300$[/tex]
For [tex]\( 5: 10 \)[/tex] and [tex]\( 100: 300 \)[/tex]:
[tex]\[ 5 \times 300 = 1500 \][/tex]
[tex]\[ 10 \times 100 = 1000 \][/tex]
Since the products are not equal, the ratios are not proportional.
Answer: no
### Pair 6: [tex]$12: 10$[/tex] and [tex]$24: 20$[/tex]
For [tex]\( 12: 10 \)[/tex] and [tex]\( 24: 20 \)[/tex]:
[tex]\[ 12 \times 20 = 240 \][/tex]
[tex]\[ 10 \times 24 = 240 \][/tex]
Since both products are equal, the ratios are proportional.
Answer: yes
So, the final answers are:
[tex]\[
\begin{array}{l|l|}
\hline \text{yes} & \text{no} \\
\hline \text{no} & \text{yes} \\
\hline \text{no} & \text{yes} \\
\end{array}
\][/tex]
