Math Assignment: Proportional Reasoning Problems Set 1 Solutions

This document provides a comprehensive solution set for a proportional reasoning assignment. The solutions cover a range of problems including determining equivalent ratios, calculating unit rates, and solving for unknowns in proportional relationships. The assignment addresses questions that require students to reason about proportions, compare ratios to determine which is greater, and apply proportional reasoning to real-world scenarios such as lemonade and candy mixtures. The solutions provided demonstrate various methods for solving these problems, including simplifying ratios, finding common denominators, and scaling up ratios without using cross-multiplication. The document also includes explanations of the reasoning behind each step, making it a valuable resource for students studying proportional reasoning.

Proportional Reasoning Problems Set 1
Respond to each of the following questions.
1. Do these ratios represent the same relationship? 3:4 and 18
24
Explain your answer.
The two ratios represent the same relationship because 3:4 is a simplified form of the
ratio 18/24.
3:4 =18:24
2. Given the rate of 6 miles every 4 hours, write 3 equivalent ratios including one-unit rate?
6
4 =12
8 = 3
2 =1.5 miles/hour
Equivalent ratios 12:8 ,3:2, 24:16
One Unit rate 1.5 miles/hour
3. Describe how you could reason about which of the following is closest to the value of n
in the equation 11.8
5.6 = n
3.9 without using cross multiplication.
A. 2.8 C. 8.2
B. 6.2 D. 10.1
8.2 The value of n has to be greater than 3.9 and its value has to be just slightly greater than
twice 3.9 (7.8). The only valid answer is thus 8.2.
4. Are these ratios equivalent? 3:5 and 6:8
Explain your answer.
The two ratios are not equal because when the ratio 6:8 is divided by 2, the resulting ratio
is 3 :4 ≠ 3 :5
5. Kerry made lemonade by adding 2 cups of lemon juice for every 3 cups of water. Jon
made lemonade by adding 4 cups of lemon juice for every 7 cups of water. Whose
lemonade will have a stronger lemon taste?
Justify your answer using proportional reasoning without using the decimals or
percentages equivalent to the ratios given.

John’s Lemon had a stronger taste because the proportion of lemon on his juice is greater
than that of Jon (2/5>4/11) as shown below
Comparing
4
11 ∧2
5
, the LCM of the denominators is 55, rewriting the proportions in
terms of denominator’s LCM, we obtain,
4
11= n
55
n= 4
11 × 55=20
2
5 = m
55
m= 2
5 × 55=22
Since m>n, Jon’s lemon had stronger lemon taste
6. Serena and Jeff have the same number of candies. Serena has 2 green candies for every 3
red candies. Jeff has 3 green candies for every 4 red candies. Who has more red candies?
Justify your answer.
Serena has more candies because she has a greater proportion of red candies as compared
to Jeff (3/5 >4/7) as shown below
When the two ratios are expressed in terms of a common denominator,
3
5 = n
35
n=3
5 ×35=21
4
7 = m
35
m= 4
7 × 35=20
n>m hence Serena had more candies
7. Describe how you could find the value of p without using cross multiplication:
2
25 = p
500
The two denominators are compared. Since they are ratios, the first step is evaluating by how
much 25 has been scaled up to arrive at 500, 500/25=20. The numerator is then scaled up by the
same value hence p=20× 2=40

8. Are these ratios equivalent? 3 m
4 u
9 m2
16 u2
Explain your answer.
The two ratios are equal because ratios can be simplified by dividing both the numerator
and denominator using the GCD. The second ratio is equivalent to the first ratio squared
thus making the two ratios equivalent.
9. Sam made juice by adding 6 cups of orange juice for every 53 cups of water. Tom made
juice by adding 5 cups of orange juice for every 42 cups of water. Which one had the
stronger juice?
Justify your answer using proportional reasoning without using the decimal or percentage
equivalent to the ratios given.
Tom because 5/47 >6/59 as shown below
5
47 = n
2773
n= 5
47 × 2773=295
6
59 = m
2773
m= 6
59 × 2773=282
n>m hence Tom had a stronger juice
10. Watson Academy boasts a ratio of 150 students to 18 teachers. How can the number of
faculty be adjusted so that the academy’s student-to-teacher ratio is 15 to 1?
Current ratio=150:18=25:3
Desired ratio=15:1
To achieve the desired ratio, the number of teachers could be reduced by 8 to 10.
Alternatively, the number of students could be increased to 270 students.