Calculating the Ratio $p:u$

The problem asks for the ratio $p:u$ given several intermediate ratios and a condition relating $u$ and $s$. We need to combine the given ratios step-by-step to find the relationship between $p$ and $s$, and then use the condition about $u$ to find $p:u$.

Combining Ratios $p:q$, $q:r$, and $r:s$

We are given:

• $p:q = 1:2$
• $q:r = 4:3$
• $r:s = 4:5$

To combine these ratios, we find a common value for the shared terms ($q$ and $r$).

1. Combine $p:q$ and $q:r$:

   We have $p:q = 1:2$ and $q:r = 4:3$. The common term is $q$. The values of $q$ are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4.

   Multiply the first ratio ($p:q$) by 2: $p:q = (1 \times 2) : (2 \times 2) = 2:4$.

   Now, $p:q = 2:4$ and $q:r = 4:3$. Since $q$ is 4 in both, we can combine them: $p:q:r = 2:4:3$.

2. Combine $p:q:r$ and $r:s$:

   We have $p:q:r = 2:4:3$ and $r:s = 4:5$. The common term is $r$. The values of $r$ are 3 and 4. The LCM of 3 and 4 is 12.

   Multiply the ratio $p:q:r$ by 4: $p:q:r = (2 \times 4) : (4 \times 4) : (3 \times 4) = 8:16:12$.

   Multiply the ratio $r:s$ by 3: $r:s = (4 \times 3) : (5 \times 3) = 12:15$.

   Now, $r$ is 12 in both combined ratios. We can combine them: $p:q:r:s = 8:16:12:15$.

This means we can represent $p, q, r, s$ as $p=8k, q=16k, r=12k, s=15k$ for some constant $k$. We are interested in $p$ and $s$.

Determining the Value of $u$

We are told that $u$ is 50% more than $s$. Mathematically, this can be written as:

$ u = s + (50\% \times s) $

$ u = s + 0.5s $

$ u = 1.5s $

Using the ratio value, $s=15k$. Therefore,

$ u = 1.5 \times (15k) $

$ u = \frac{3}{2} \times 15k $

$ u = \frac{45}{2}k $

$ u = 22.5k $

Calculating the Final Ratio $p:u$

We need to find the ratio $p:u$. We know:

• $p = 8k$
• $u = 22.5k$

The ratio $p:u$ is:

$ p:u = 8k : 22.5k $

We can cancel $k$ from both sides:

$ p:u = 8 : 22.5 $

To express this ratio using integers, we can multiply both sides by 2 to eliminate the decimal:

$ p:u = (8 \times 2) : (22.5 \times 2) $

$ p:u = 16 : 45 $

Final Answer

The ratio $p:u$ is $16:45$.