Question

Elyse and Jamir each make lemonade. Elyse mixes 1
7
3
​
cups of water with
2
1
​
cups of lemonade concentrate. Jamir mixes 1
3
1
​
cups of water with
5
2
​
cups of lemonade concentrate. Which lemonade has a more lemony taste?
To divide
2
1
​
by 1
7
3
​
, start by writing 1
7
3
​
as a fraction greater than 1.

Elyse's Lemonade Recipe
÷ 1
7
3
​

÷ 1
7
3
​

Water
(cups) Lemonade Concentrate
(cups)
1
7
3
​

2
1
​

1 ?
2
1
​
÷ 1
7
3
​

=
2
1
​
÷
10
7

Next, rewrite
2
1
​
÷
7
10
​
as a multiplication expression.

Elyse's Lemonade Recipe
÷ 1
7
3
​

÷ 1
7
3
​

Water
(cups) Lemonade Concentrate
(cups)
1
7
3
​

2
1
​

1 ?

2
1
​
÷ 1
7
3
​


=
2
1
​
÷
10
7

=
2
1
​
•
7
10

To continue finding the number of cups of lemonade concentrate per 1 cup of water in Elyse’s recipe, multiply
2
1
​
•
10
7
​
. Start by writing
2
1
​
•
10
7
​
as a single fraction.

Elyse's Lemonade Recipe
÷ 1
7
3
​

÷ 1
7
3
​

Water
(cups) Lemonade Concentrate
(cups)
1
7
3
​

2
1
​

1 ?

2
1
​
÷ 1
7
3
​


=
2
1
​
÷
10
7

=
2
1
​
•
7
10

=
7
•
1
2
•
10

Multiply in the numerator and denominator to solve, then record your answer in the table to show the number of cups of lemonade concentrate Elyse needs for 1 cup of water.

Elyse's Lemonade Recipe
÷ 1
7
3
​

÷ 1
7
3
​

Water
(cups) Lemonade Concentrate
(cups)
1
7
3
​

2
1
​

1

2
1
​
÷ 1
7
3
​


=
2
1
​
÷
10
7

=
2
1
​
•
7
10

=
1 • 7
2 • 10

=

Answer 1

To find out how much lemonade concentrate Elyse needs for 1 cup of water, let's go through the calculations step-by-step.

### Step 1: Convert Elyse's recipe measurements to improper fractions.

Elyse mixes:
- \( 1 \frac{7}{3} \) cups of water
- \( 2 \frac{1}{2} \) cups of lemonade concentrate

Convert these to improper fractions:

1. For water:
\[
1 \frac{7}{3} = \frac{3 \times 1 + 7}{3} = \frac{3 + 7}{3} = \frac{10}{3}
\]

2. For lemonade concentrate:
\[
2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}
\]

### Step 2: Calculate how much lemonade concentrate is needed per 1 cup of water.

Now we need to divide the amount of lemonade concentrate by the amount of water:
\[
\text{Lemonade concentrate per cup of water} = \frac{\frac{5}{2}}{\frac{10}{3}}
\]

### Step 3: Rewrite the division as multiplication by the reciprocal.

This can be rewritten as:
\[
\frac{5}{2} \times \frac{3}{10}
\]

### Step 4: Multiply the fractions.

Now, multiply the numerators and the denominators:
\[
\frac{5 \times 3}{2 \times 10} = \frac{15}{20}
\]

### Step 5: Simplify the fraction.

To simplify \( \frac{15}{20} \):
\[
\frac{15 \div 5}{20 \div 5} = \frac{3}{4}
\]

This means Elyse needs \( \frac{3}{4} \) cups of lemonade concentrate for 1 cup of water.

### Summary Table Entry for Elyse:
Elyse's Lemonade Recipe:
- Water (cups): \( 1 \frac{7}{3} \)
- Lemonade Concentrate (cups): \( 2 \frac{1}{2} \)
- Concentrate needed per 1 cup of water: \( \frac{3}{4} \)

---

### Now for Jamir’s lemonade:

We will do the same for Jamir, who mixes:
- \( 1 \frac{3}{1} \) cups of water, which is simply \( 4 \) cups of water (1 is included in the fraction)
- \( 5 \frac{2}{1} \) cups of lemonade concentrate, which is \( 7 \) cups of concentrate.

### Convert measurements to improper fractions:
- Water: \( 1 \frac{3}{1} = 4 \)
- Lemonade Concentrate: \( 5 \frac{2}{1} = 7 \)

### Calculate how much lemonade concentrate is needed per 1 cup of water:

Now we divide the amount of concentrate by the amount of water:
\[
\text{Lemonade concentrate per cup of water} = \frac{7}{4}
\]

This means Jamir needs \( \frac{7}{4} \) cups of lemonade concentrate for 1 cup of water.

### Summary Table Entry for Jamir:
Jamir's Lemonade Recipe:
- Water (cups): \( 4 \)
- Lemonade Concentrate (cups): \( 7 \)
- Concentrate needed per 1 cup of water: \( \frac{7}{4} \)

### Conclusion:

- Elyse’s lemonade concentration per 1 cup of water: \( \frac{3}{4} \)
- Jamir’s lemonade concentration per 1 cup of water: \( \frac{7}{4} \)

**Comparison**:
Jamir's lemonade has a more lemony taste because it has a higher concentration of lemonade concentrate per cup of water.