Question

8. Joe can chop vegetables in 5 minutes, and Rich can chop the same amount of vegetables in 4 minutes. Working together, how long will it take them to chop that batch of vegetables? 9. At Ricardo's Tacos, four tacos and two orders of chips cost the same as two tacos and four orders of chips. If Ricardo's charges $2.00 for a single order of chips, how much does Ricardo's charge for 1 taco? 10. Broccoli is $1.69 per pound. Meg paid $8.45 for broccoli. How many pounds did she purchase?

Answer 1

Answer:

• Question 8: 2.22 minutes

• Question 9: $2.00 for one taco

• Question 10: 5.00 pounds

Explanation:

8. Joe can chop vegetables in 5 minutes, and Rich can chop the same amount of vegetables in 4 minutes. Working together, how long will it take them to chop that batch of vegetables?

Name v the amount of vegetables

• Joe can chop that amount is 5 minutes, then his speed is v/5 (vegetables per minute).

• Rich can chop the same amount of vegetables in 4 minutes, then his speed is v/4 (vegetables per minute)

Working together, the combined speed is the sum of the two speeds: v/5 + v/4

Thus, the speed working together is:

            $\frac{v}{5} +\frac{v}{4}=\frac{4v+5v}{20}=\frac{9v}{20}$

Hence, they can chop 9 times the given amount of vegetables (v) in 20 minutes.

And the time to chop the given amount of vegetables (v) is 20 divided by 9.

               $time=amount/speed\\\\time=v/(9v/20)\\\\time=20v/(9v)=20/9=2.22$

That is 2.22 minutes to chop all the vegetables working together.

9. At Ricardo's Tacos, four tacos and two orders of chips cost the same as two tacos and four orders of chips. If Ricardo's charges $2.00 for a single order of chips, how much does Ricardo's charge for 1 taco?

Use T for the cost of tacos and C for the cost of orders of chips

• Cost of four tacos and two orders of chips: 4T + 2C

• Cost of two tacos and four order of chips: 2T + 4C

Ricardo's charges the same for those orders:

• 4T + 2C = 2T + 4C

Ricardo's charges $2.00 for a single order of chips:

• C = 2

Substitute C = 2 in 4T + 2C = 2T + 4C and solve:

   Substitution:

• 4T + 2(2) = 2T + 4(2)

   Do the operations:

• 4T + 4 = 2T + 8

   Subtract 4 from both sides

• 4T = 2T + 4

   Subtract 2T from both sides

• 4T - 2T = 4

   Combine like terms

• 2T = 4

   Divide both sides by 2

• T = 2

Hence, Ricardo's charges $2.00 for one taco.

10. Broccoli is $1.69 per pound. Meg paid $8.45 for broccoli. How many pounds did she purchase?

You must divide the amount paid ($8.45) by the unit price ($1.69/lb)

        $\ 8.45/(\ 1.69/lb)=5.00lb$

In the operation, $ appears both in the numerator and the denominator so they cancel out each other. The unit pounds (lb) appears dividing the denominator, thus it passes to the numerator.

Hence, Meg purchased 5.00 pounds