SHSAT: Tables and proportional relationships
Test whether a table is proportional, extend it with a constant of proportionality, read cumulative tables, and compare rates fairly.
How can you tell whether a table shows a proportional relationship, and how do you use it once you know?
Short answer
Divide y by x in every row. If the result is always the same number k, the rule is y = kx; use it to find any missing value.
Know first: Dividing and simplifying fractions, Converting fractions to percents
Step 1 of 5
Test a table before you extend it
A relationship is proportional when y ÷ x gives the same number in every row. That number is the constant of proportionality, and the rule is y = kx. Its graph is a line through (0, 0).
Many tables go up by equal steps without being proportional. A fixed starting amount, such as a delivery fee, breaks the pattern. Divide each y by its x to check.
| Table type | How to recognize it | How to use it |
|---|---|---|
| Proportional | y ÷ x is the same in every row | y = kx; to find x, divide y by k |
| Linear with a starting amount | equal steps, but y ÷ x changes | find the step per unit and the starting amount |
| Cumulative (running total) | each row includes all earlier amounts | subtract two rows to get the amount in between |
Read the table's title and caption. Words such as total so far signal a cumulative table.
Step 2 of 5
Find k, then use it
Worked example: Extend a proportional table
A table shows a proportional relationship with rows (3, 12), (5, 20) and (8, 32). What is y when x = 12?
- A3
- B36
- C48
- D144
- 1
Check each ratio
12 ÷ 3 = 4, 20 ÷ 5 = 4, 32 ÷ 8 = 4. The constant is k = 4.
- 2
Write the rule
y = 4x.
- 3
Use it
y = 4 × 12 = 48.
- 4
See why the other choices are there
3 divides x by k. 36 adds the change in x, 4, to the last y. 144 multiplies 12 by 12, the first y value, instead of by the rate.
Answer
y = 48
Wrong: The table (1, 5), (2, 8), (3, 11) goes up 3 each time, so y = 3x, and at x = 10, y = 30.
Right: y ÷ x is 5, then 4, then 11/3, so the table is not proportional. The rule is y = 3x + 2, and at x = 10, y = 32.
Equal steps show a linear pattern. Only a constant ratio y ÷ x shows a proportional one.
Step 3 of 5
Use the table the way it was built
Check yourself · Question 1
A cumulative table shows the total pages a student has read by the end of each day: Monday 45, Tuesday 83, Wednesday 120. How many pages did the student read on Wednesday?
Answer: A
Each total includes the earlier days. Wednesday's pages are 120 − 83 = 37.
Check yourself · Question 2
Brand A sells 5 cans for $6.50. Brand B sells 8 cans for $10.00. How many cents less per can does Brand B charge?
Answer: B
Brand A: 650 cents ÷ 5 = 130 cents per can. Brand B: 1000 cents ÷ 8 = 125 cents per can. B charges 130 − 125 = 5 cents less.
Trap answer: 350
Comparing packages of different sizes
- Why it looks right
- The two prices are right there in the question, and subtracting them is quick.
- Why it's wrong
- The packages hold 5 and 8 cans, so their prices can't be compared directly. Compare the price for one can.
- The right answer
- 5 cents, from 130 − 125.
Check yourself · Question 3
Class A had 18 of 24 students pass a test. Class B had 32 of 40 pass. By how many percentage points does Class B's pass rate exceed Class A's?
Answer: A
Class A: 18 ÷ 24 = 75%. Class B: 32 ÷ 40 = 80%. The difference is 80 − 75 = 5 percentage points.
Common mistake
I read a running total as the amount for that one period.
- Why it's tempting
- Each row sits next to a single time, so it looks like that time's amount.
- Do this instead
- Check the title or caption for words like cumulative or total so far. If they're there, subtract the row before to get one period's amount.
Step 4 of 5
What to remember
Remember
- A table is proportional only if y ÷ x is the same in every row. Then y = kx.
- Cumulative tables need subtraction to find the amount in one interval.
- Compare unit prices, not package prices, and read whether a question wants percentage points or a percent change.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what did you get when you divided y by x in each row, and how did you use that number?