Question

Two packets of drink mix should be mixed with 3 quarts of water. Use the double number line to find how many packets of drink mix to mix with 54 quarts of water. A. 30 packets B. 33 packets C. 36 packets D. 39 packets

Short answer

36 packets

Step-by-step solution

- Understand the Ratio

First, identify the given ratio of drink mix to water. Two packets correspond to three quarts of water.

- Set Up the Double Number Line

Construct a double number line with one line representing packets of drink mix and the other line representing quarts of water. Start by marking 2 packets on one line and 3 quarts on the corresponding position on the other line.

- Scale Up the Ratio

To find how many packets are needed for 54 quarts of water, scale up the number line accordingly. Since 2 packets correspond to 3 quarts, divide 54 by 3 to find how many times the ratio is scaled.

- Perform the Division

Divide 54 by 3 to determine the multiplier: 54 ÷ 3 = 18. This multiplier tells us how many times the initial ratio of 2 packets per 3 quarts fits into 54 quarts.

- Calculate the Number of Packets

Multiply the number of packets by the multiplier found in the previous step: 2 packets × 18 = 36 packets.

- Verify the Answer

Verify that with 54 quarts and the solution of 36 packets, the ratio remains consistent. 18 times the initial ratio of 2 packets to 3 quarts confirms the calculation.

Key concepts

ratios and proportions

Ratios and proportions are key concepts in mathematical reasoning, especially in the GED test. A ratio is a comparison of two quantities that shows the relative size of one quantity to another. For this exercise, the ratio is 2 packets of drink mix to 3 quarts of water.

Proportion is an equation that states that two ratios are equal. When we scale up or down the quantities, we are maintaining the same proportion. To ensure the quantities stay proportional, you multiply or divide both parts of the ratio by the same number.

In our example, we begin with the ratio 2 packets to 3 quarts. We need to determine how many packets would be required for 54 quarts of water while preserving the ratio.

double number line

A double number line is a visual aid that helps in solving ratio and proportion problems. It consists of two parallel lines: one for each part of the ratio. This method helps in understanding the relationship between the two quantities more clearly.

Start by marking the initial ratio on the lines. Here, the line for packets starts at 2 and the line for quarts starts at 3. Next, extend both lines while keeping the same intervals to scale up the quantities. This way, you can see visually how many packets are needed for increasing amounts of water.

For 54 quarts of water, you divide 54 by 3 (the original quarts in the ratio), to determine the scaling factor. After finding this scaling factor, you apply it to 2 packets (the original packets in the ratio).

problem-solving steps

Solving ratio problems involves clear and sequential steps. Let's outline the steps for this specific exercise:

• Identify the initial ratio (2 packets: 3 quarts).
• Set up the double number line with the initial values of packets and quarts.
• Determine how many times the desired quantity (54 quarts) fits into the smaller quantity (3 quarts).
• Calculate the scaling factor by dividing 54 by 3, which gives us 18.
• Multiply the number of packets (2) by the scaling factor (18).
• Verify the answer by checking if 36 packets to 54 quarts maintains the ratio.

Following these problem-solving steps helps simplify even seemingly complex ratio problems.

scaling up ratios

Scaling up ratios means increasing the quantities in the ratio while maintaining the same proportion. This is achieved by multiplying both parts of the ratio by the same number.

In our scenario, we need to scale up from 3 quarts to 54 quarts. To do this, we first determine the scaling factor by dividing 54 by 3, which equals 18. This tells us that we need to multiply both parts of the ratio by 18 to maintain the proportion.

Therefore, we take the initial packets (2) and multiply by 18, yielding 36 packets. This method ensures the proportion of 2 packets for every 3 quarts is preserved, just on a larger scale.

This technique is useful not only in everyday tasks like recipes but also in solving various mathematical problems efficiently.