Lester and ralph work at an athletic shoe store in the mall, but whenever they goof off too much, their boss punishes them by making them lace-up all the inventory. If it takes lester 30 seconds to lace up a high top and ralph 35 seconds to do the same, how many seconds will it take them to lace up 260 high tops

Answer 1

To find out how long it will take Lester and Ralph to lace up 260 high tops, we first determine their combined lacing rate.

1. Lester's rate: Lester takes 30 seconds to lace up one high top. Therefore, his rate is: \[ \frac{1 \text{ high top}}{30 \text{ seconds}} \quad \text{or} \quad \frac{2 \text{ high tops}}{60 \text{ seconds}} \quad \text{or} \quad \frac{3 \text{ high tops}}{90 \text{ seconds}} \quad \text{...} \]

2. Ralph's rate: Ralph takes 35 seconds to lace up one high top. So, his rate is: \[ \frac{1 \text{ high top}}{35 \text{ seconds}} \]

3. Combined rate: To find their combined rate when working together, we add their individual rates: \[ \text{Combined rate} = \frac{1}{30} + \frac{1}{35} \]

   To add these fractions, we find a common denominator. The least common multiple (LCM) of 30 and 35 is 210.

   Converting the rates to have a common denominator: \[ \frac{1}{30} = \frac{7}{210} \quad \text{and} \quad \frac{1}{35} = \frac{6}{210} \]

   So, the combined rate is: \[ \text{Combined rate} = \frac{7}{210} + \frac{6}{210} = \frac{13}{210} \]

4. Time to lace up one high top: The time needed to lace up one high top is the reciprocal of the combined rate: \[ \text{Time for 1 high top} = \frac{210}{13} \text{ seconds} \]

5. Total time for 260 high tops: To find the total time for 260 high tops, we multiply the time for one high top by 260: \[ \text{Total time} = 260 \times \frac{210}{13} \]

   Calculating that: \[ 260 \div 13 = 20 \] So, \[ 20 \times 210 = 4200 \text{ seconds} \]

Thus, it will take Lester and Ralph 4200 seconds to lace up 260 high tops.