Question
GIA’S BAG OF SNACK MIX
| Snack | Number in Bag |
|---|---|
| Pretzels | 12 |
| Raisins | 9 |
Gia and her friends counted the numbers of pretzels and raisins in their bags of snack mix. The table above shows the numbers in Gia’s bag. Which other snack mix bag has a ratio of pretzels to raisins in the same proportional relationship as Gia’s bag?
- A
ADELE’S BAG OF SNACK MIX
Snack Number in Bag Pretzels 6 Raisins 18 - B
CARL’S BAG OF SNACK MIX
Snack Number in Bag Pretzels 15 Raisins 12 - C
ISABEL’S BAG OF SNACK MIX
Snack Number in Bag Pretzels 18 Raisins 15 - D
TREVON’S BAG OF SNACK MIX
Snack Number in Bag Pretzels 16 Raisins 12
Show the answer
Correct answer: D
The ratio of pretzels to raisins in Gia’s bag of snack mix, , is equivalent to the ratio of pretzels to raisins in Trevon’s bag, .
Question analysis
Gia's bag of snack mix holds 12 pretzels and 9 raisins, and four friends' bags are shown in tables of their own. Students choose the bag whose pretzel-to-raisin ratio is in the same proportional relationship as Gia's, a test of equivalent ratios where bags that look similar at a glance need an exact check.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Grade 7 students regularly decide whether two ratios are equivalent. Every count here is a small whole number that reduces quickly, so the task is routine apart from having to check four bags one by one.
- Estimated difficulty
- Straightforward: two steps or one careful reading.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 3, 96 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Every count is small and reduces quickly, so the question turns on whether students compare ratios or differences. Gia has 3 more pretzels than raisins, and so do Carl and Isabel, so a student comparing differences finds two apparent matches and has nothing to choose between them. Reducing each bag to lowest terms, Gia's 12:9 to 4:3, leaves exactly one match and exposes the additive pull.
Every choice must be checked minor Choosing the answer
Each friend's bag has to be reduced in turn, 6:18 to 1:3, 15:12 to 5:4, 18:15 to 6:5 and 16:12 to 4:3, since none can be ruled out at a glance. (Embretson & Wetzel, 1987)
Reading a figure, table or graph minor Understanding the question
The counts sit in five small tables, one per bag, and pretzels must stay first in every ratio to match Gia's 12 pretzels to 9 raisins. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Strategy: Find Gia's pretzels per raisin as a fraction, then compute the same quotient for each friend's bag, keeping pretzels on top, and look for an exact match.
Traps
- Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
The wrong answers, classified
A Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Halves Gia's 12 pretzels to 6 and doubles her 9 raisins to 18, as if one change balanced the other, but 6:18 reduces to 1:3.
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Adds 3 to each of Gia's counts, giving 15 and 12, which keeps her gap of 3 pretzels over raisins instead of her ratio; 15:12 reduces to 5:4.
C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Adds 6 to each count, giving 18 and 15, so pretzels still lead raisins by 3, but 18:15 reduces to 6:5, not Gia's 4:3.
Worked solution, step by step
Formulas used
- Equivalent ratios
- a and b are the two amounts being compared, in order. k is any number that divides both of them evenly. Dividing both amounts by the same number keeps the ratio the same. When only 1 divides both, the ratio is in simplest form.
Sort out the question
A ratio compares two amounts. Gia’s bag has 12 pretzels and 9 raisins. So her ratio of pretzels to raisins is 12 : 9.
The same proportional relationship means the ratios are equal once both are simplified. Keep pretzels first and raisins second every time.
Simplify Gia’s ratio
To simplify a ratio, divide both numbers by the same number. Both 12 and 9 divide evenly by 3.
Gia’s bag has 4 pretzels for every 3 raisins.
Simplify each friend’s bag
Do the same for each choice. Divide both numbers by the biggest number that goes into both.
Choice Pretzels : raisins Divide by Simplest form A, Adele 6 : 18 6 1 : 3 B, Carl 15 : 12 3 5 : 4 C, Isabel 18 : 15 3 6 : 5 D, Trevon 16 : 12 4 4 : 3 Pick the matching bag
Only Trevon’s bag simplifies to 4 : 3, just like Gia’s.
Watch out: Carl’s and Isabel’s bags each have 3 more pretzels than raisins, the same gap as Gia’s. But adding the same number to both parts changes the ratio. That trap leads to choice B or C.
So the answer is D, Trevon’s bag.
Check your answer
Go the other way. Multiply both parts of 4 : 3 by 4 to get 16 : 12. Those are Trevon’s exact counts. Or divide pretzels by raisins: and both equal .
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