Step 1: Define the tex count system. A tex count of 30 means 1000 meters of yarn weigh 30 grams.
\[ 30 \, \text{tex} = \frac{30 \, \text{grams}}{1000 \, \text{meters}} \]
Step 2: Set up the problem. The total yarn weight is 75 grams, and the weight per 1000 meters is known. Find the total length.
Express the relationship as:\[ \text{Length (m)} = \frac{\text{Total Weight (g)}}{\text{Weight per meter (g/m)}} \]First, find the weight per meter:\[ \text{Weight per meter} = \frac{30 \, \text{g}}{1000 \, \text{m}} = 0.03 \, \text{g/m} \]
Step 3: Calculate the total length.
\[ \text{Length (m)} = \frac{75 \, \text{g}}{0.03 \, \text{g/m}} = \frac{75}{3/100} = 75 \times \frac{100}{3} = 25 \times 100 = 2500 \, \text{meters} \]Alternative Method (using proportion):If 30 grams corresponds to 1000 meters,Then 75 grams corresponds to X meters.\[ \frac{1000 \, \text{m}}{30 \, \text{g}} = \frac{X \, \text{m}}{75 \, \text{g}} \]\[ X = \frac{1000 \times 75}{30} = \frac{75000}{30} = \frac{7500}{3} = 2500 \, \text{meters} \]