Ratio Problem Introduction

This problem involves understanding and combining multiple ratios to find a final desired ratio. We are given several relationships between variables $p\$,$ q\$,$ r\$,$ s\$,\; and$ u\$.$$$$$

Our goal is to determine the ratio of $p\$\; to$ u\$,\; which\; is\; expressed\; as$ p:u\$.$$$

Given Ratio Information

Let's list down all the given ratios and the additional relationship for variable $u\$:$

• The ratio of $p\$\; to$ q\$\; is$ 1:2\$.\; This\; can\; be\; written\; mathematically\; as$ {\displaystyle p/q=1/2}\$.$$$$
• The ratio of $q\$\; to$ r\$\; is$ 4:3\$.\; This\; means$ {\displaystyle q/r=4/3}\$.$$$$
• The ratio of $r\$\; to$ s\$\; is$ 4:5\$.\; This\; means$ {\displaystyle r/s=4/5}\$.$$$$
• The variable $u\$\; is$ 50\%\$\; more\; than$ s\$.\; This\; can\; be\; translated\; into\; the\; equation$ u=s+0.50s=1.5s\$,\; or\; equivalently,\; as\; a\; fraction,$ {\displaystyle u=\frac{3}{2}s}\$.$$$$$

Combining Ratios: Step-by-Step

To find the ratio $p:u\$,\; our\; first\; essential\; step\; is\; to\; combine\; the\; individual\; ratios\; into\; a\; single\; continuous\; ratio\; for$ p:q:r:s\$.$$

Step 1: Combine $p:q\$\; and$ q:r\$$$

We are given the following ratios:

• $p:q=1:2\$$
• $q:r=4:3\$$

To combine these ratios, the value representing $q\$\; must\; be\; consistent.\; In\; the\; first\; ratio,$ q\$\; is$ 2\$,\; and\; in\; the\; second,\; it\; is$ 4\$.\; The\; Least\; Common\; Multiple\; (LCM)\; of$ 2\$\; and$ 4\$\; is$ 4\$.$$$$$$$

To make the $q\$\; value$ 4\$\; in\; the\; first\; ratio,\; we\; multiply\; both\; parts\; of$ p:q\$\; by$ 2\$:$$$$

$p:q=(1\times 2):(2\times 2)=2:4\$$

Now that the $q\$\; values\; are\; consistent,\; we\; can\; write\; the\; combined\; ratio:$

$p:q:r=2:4:3\$$

Step 2: Combine $p:q:r\$\; and$ r:s\$$$

We now have:

• $p:q:r=2:4:3\$$
• $r:s=4:5\$$

The common variable here is $r\$.\; The\; values\; for$ r\$\; are$ 3\$\; and$ 4\$.\; The\; LCM\; of$ 3\$\; and$ 4\$\; is$ 12\$.$$$$$$$

To make the $r\$\; value$ 12\$\; in\; the\; first\; combined\; ratio$ p:q:r\$,\; we\; multiply\; all\; parts\; by$ 4\$:$$$$

$p:q:r=(2\times 4):(4\times 4):(3\times 4)=8:16:12\$$

To make the $r\$\; value$ 12\$\; in\; the\; ratio$ r:s\$,\; we\; multiply\; both\; parts\; by$ 3\$:$$$$

$r:s=(4\times 3):(5\times 3)=12:15\$$

Now that all common terms are consistent, we can combine all four variables into a single continuous ratio:

$p:q:r:s=8:16:12:15\$$

Calculating the Ratio p ∶ u

From our combined ratio $p:q:r:s=8:16:12:15\$,\; we\; can\; express$ p\$\; and$ s\$\; in\; terms\; of\; a\; common\; constant$ k\$.\; Let$ p=8k\$\; and$ s=15k\$.$$$$$$

We are also given that $u\$\; is$ 50\%\$\; more\; than$ s\$.\; This\; means:$$$

$$u=s+50\%\text{of}s\$$$

$$u=s+\frac{50}{100}s\$$$

$$u=s+\frac{1}{2}s\$$$

$$u=\frac{3}{2}s\$$$

Now, substitute the value of $s\$\; from\; the\; combined\; ratio\; into\; the\; equation\; for$ u\$:$$

$$u=\frac{3}{2}\left(15k\right)\$$$

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

Finally, we can determine the ratio $p:u\$:$

$$p:u=8k:\frac{45}{2}k\$$$

We can divide both sides of the ratio by $k\$\; (assuming$ k\$\; is\; not\; zero):$$

$$p:u=8:452\$$$

To eliminate the fraction in the ratio, multiply both sides of the ratio by $2\$:$

$$p:u=(8\times 2):(\frac{45}{2}\times 2)\$$$

$$p:u=16:45\$$$

Final Answer

The final ratio of $p\$\; to$ u\$\; is$ 16:45\$.$$$