Question

A concrete mixing tub contains 34 liters of water and 22 liters of sand. Starting at the same time, water is added to the tub at a rate of 5 liters per minute, while sand is also added to the tub at a rate of 7 liters per minute.

Part A
In how many minutes will the tub contain equal amounts of water and sand?

2 minutes
6 minutes
12 minutes
24 minutes

Part B
When the amounts of water and sand in the tub are equal, how many liters of each will be in there?

64 liters
76 liters
154 liters
170 liters

Answer 1

To solve this problem, let’s first define the initial quantities and the rates at which water and sand are added.

**Initial amounts:**
- Water: 34 liters
- Sand: 22 liters

**Rates of addition:**
- Water is added at a rate of 5 liters per minute.
- Sand is added at a rate of 7 liters per minute.

Let \( t \) be the time in minutes after starting the addition of water and sand.

**Amounts after \( t \) minutes:**
- Amount of water after \( t \) minutes: \( W(t) = 34 + 5t \)
- Amount of sand after \( t \) minutes: \( S(t) = 22 + 7t \)

**Part A: Finding when both amounts are equal**

We want to find \( t \) when \( W(t) = S(t) \):
\[
34 + 5t = 22 + 7t
\]

Now, let’s solve the equation:

1. Rearrange the equation:
\[
34 - 22 = 7t - 5t
\]
\[
12 = 2t
\]
2. Divide both sides by 2:
\[
t = 6
\]

So, the tub will contain equal amounts of water and sand after **6 minutes**.

**Part B: Finding the amounts when they are equal**

Now we need to find how much water and sand is in the tub after 6 minutes:

Substituting \( t = 6 \) into the equations for \( W(t) \) and \( S(t) \):

- Amount of water:
\[
W(6) = 34 + 5 \times 6 = 34 + 30 = 64 \text{ liters}
\]

- Amount of sand:
\[
S(6) = 22 + 7 \times 6 = 22 + 42 = 64 \text{ liters}
\]

When the amounts are equal, there will be **64 liters** of both water and sand.

**Final answers:**
- **Part A:** 6 minutes
- **Part B:** 64 liters