We are given the ratio of red to clay particles is \( 5:3 \), and there are 15 red particles. Let the number of clay particles be \( c \). Set up the proportion:
["Understanding Particle Ratios: Solving for Clay Particles Using Proportions", "In many scientific and geological analyses, understanding the ratio of different types of particles helps describe soil composition, sediment mixtures, or material blends. One common ratio given is that of red particles to clay particles, expressed as ( 5:3 ). When exactly 15 red particles are known, determining the number of clay particles becomes a straightforward application of proportional reasoning.", "### Setting Up the Proportion", "The ratio ( 5:3 ) means that for every 5 red particles, there are 3 clay particles. This relationship can be expressed mathematically using a proportion. Let ( c ) represent the unknown number of clay particles. The proportion set up from the ratio is:", "[\n\frac{5}{3} = \frac{15}{c}\n]", "Here, 5 corresponds to the red particle count, 3 to clay particles, 15 is the known red particle quantity, and ( c ) is the quantity we solve for.", "### Solving the Proportion", "To find ( c ), cross-multiply:", "[\n5 \cdot c = 3 \cdot 15\n]", "[\n5c = 45\n]", "Divide both sides by 5:", "[\nc = \frac{45}{5} = 9\n]", "Thus, there are ( 9 ) clay particles when the red to clay particle ratio is ( 5:3 ) and there are 15 red particles.", "### Conclusion", "Setting up the correct proportion from the given ratio and known quantity allows us to easily solve for unknowns using basic algebra. This method applies in environmental science, geology, and material science, where particle ratios describe composition and influence material behavior.", "If you're analyzing sediment samples or designing composite materials, understanding and setting up proportions like ( \frac{5}{3} = \frac{15}{c} ) provides a clear path to solutions.", "---", "Key takeaway: When given a particle ratio and a known quantity, forming the proportion ( \frac{a}{b} = \frac{r}{c} ), then solving for ( c ) via cross-multiplication enables quick and accurate determination of unknown particle counts."]
