Question
GIA’S MIX
| Snack | Number in Bag |
|---|---|
| Pretzels | 12 |
| Raisins | 9 |
Gia and her friends counted the numbers of pretzels and raisins in their snack mix. The table above shows the numbers in Gia’s mix. Which other mix has a ratio of pretzels to raisins in the same proportional relationship as Gia’s mix?
- A
ADELE’S MIX
Snack Number in Bag Pretzels 6 Raisins 18 - B
CARL’S MIX
Snack Number in Bag Pretzels 15 Raisins 12 - C
ISABEL’S MIX
Snack Number in Bag Pretzels 18 Raisins 15 - D
TREVON’S 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 snack mix table lists how many pretzels and raisins are in her bag, and four friends' tables list theirs. Students pick the mix whose ratio of pretzels to raisins matches Gia's. It tests equivalent ratios and proportional relationships, and some mixes look close at a glance, so each ratio has to be tested exactly.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Testing whether two ratios are equivalent is core grade 7 proportional reasoning. The numbers are small whole numbers in four short tables, so the work is routine, though every table has to be checked against Gia's.
- 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 4, 77 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
A short ratio check made harder only by an additive pull: Carl's and Isabel's mixes keep the gap of 3 between pretzels and raisins that Gia's mix has, so students who compare differences instead of ratios see two matches. Reducing every table to lowest terms, Gia's 12:9 to 4:3, shows that exactly one mix keeps the ratio.
Every choice must be checked minor Choosing the answer
Each of the four tables has to be reduced in turn, to 1:3, 5:4, 6:5 and 4:3, because none of them 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 separate tables, and pretzels must stay first in every ratio, in the same order as Gia's 12 pretzels to 9 raisins. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Strategy: Simplify the pretzels-to-raisins ratio in Gia's table to lowest terms, then simplify each other mix's ratio in the same order and compare.
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 and doubles her 9 raisins, which keeps the product 108 the same but changes the ratio from 4:3 to 1:3.
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Adds 3 to both of Gia's counts, 12 + 3 and 9 + 3, keeping the difference of 3 instead of the 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 both counts, giving 18 and 15, so pretzels still outnumber raisins by 3, but 18:15 reduces to 6:5, not 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 evenly. Dividing both amounts by the same number keeps the ratio the same. When no number but 1 divides both, the ratio is in simplest form.
Sort out the question
A ratio compares two amounts. Gia’s mix 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 you simplify them. For example, 2 : 1 and 6 : 3 are equal. Always keep pretzels first and raisins second.
Simplify Gia’s ratio
Both 12 and 9 can be divided by 3. So divide both by 3.
Gia’s mix has 4 pretzels for every 3 raisins.
Simplify each other mix
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 mix
Only Trevon’s mix simplifies to 4 : 3, the same as Gia’s.
Watch out: Carl’s and Isabel’s mixes each have 3 more pretzels than raisins, just like Gia’s. But adding the same number to both amounts changes the ratio. Equal ratios come from multiplying or dividing. That trap leads to choice B or C.
So the answer is D, Trevon’s mix.
Check your answer
Go the other way. Multiply both parts of 4 : 3 by 4. You get 16 : 12, which are Trevon’s exact numbers. You can also divide pretzels by raisins. Both and equal .
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