SAT Math Review · Problem-Solving and Data Analysis
Data word problems with the calculator
These items are not clever. They are fiddly — three numbers pulled out of a table, one expression, one careful press of the equals key.
To the nearest percent, what percent of the items sold were online sales that were later returned?
Of the 620 items a store sold last month, 248 were sold online. Of those online sales, 93 were later returned.
“What percent of the items sold” names the denominator: all 620. So 93620 = 0.15 = 15%. C uses the online subtotal, 93248 = 37.5%, which answers “what percent of the online sales were returned” — a true statement about a different quantity. B uses the other subtotal, the 620 − 248 = 372 in-store sales, giving 93372 = 25%. D never uses the 93 at all: 248620 = 40% is the online share of sales, returned or not. Every wrong choice here is a correct calculation of something the question did not ask for.
1
Write the whole expression before you compute anything.
Read the last sentence first to find the quantity asked for. Pull out only the numbers that quantity needs. Write them as one expression. Then compute, once.
1. what is being asked for?2. which numbers feed it?3. write the expression4. compute once, round last
asked for: returned online items ÷ all items
the opening problem, read backwards
needed: 93 and 620 — not 248
a stem often hands you a number you do not need
93620 × 100 = 15%
one expression, one calculation
Computing as you read is how the mark is lost
Students who divide 93 by 248 the moment they finish that sentence have committed to a denominator before knowing which one the question wants. Nothing in this lesson is difficult; almost everything in it is a bookkeeping error waiting to happen. The expression on paper is what stops it.
2
In a two-way table, the question names the denominator.
Each row is a grade level, each column a way of getting to school. Only the margins are totals; every inner cell counts students in both categories.
Grade
Walk
Bike
Bus
Car
Total
9
34
27
58
19
138
10
29
22
61
26
138
11
21
15
47
43
126
12
18
11
32
57
118
Total
102
75
198
145
520
“of the bus riders, what percent are in grade 10?”
bottom = the Bus column total, 198
61198 = 0.3081 → 30.8%
the cell where row 10 meets column Bus
“what percent of grade 12 students ride in a car?”
bottom = the grade-12 row total, 118
57118 = 0.4831 → 48.3%
same cell, different question, different total
“what percent of all students bike to school?”
bottom = the grand total, 520
75520 = 0.1442 → 14.4%
the Bike column total over everything
Three totals sit next to every cell
The 57 grade-12 car riders can be divided by 118 (their row), by 145 (their column) or by 520 (everyone), giving 48.3%, 39.3% and 11.0%. All three appear as choices. Circle the total the sentence names before you touch the calculator — “of the grade 12 students” is the row, “of the students who ride in a car” is the column, “of all students surveyed” is the corner.
3
A mean is a total in disguise: sum = mean × count.
Whenever a question changes a data set — adds a value, removes one, asks what score is needed — convert every mean into a sum first. Sums add; means do not.
mean = sumcount → sum = mean × count
the only formula this whole step needs
7 readings, mean 12.4 → sum = 86.8
7 × 12.4, the total those readings represent
add an eighth reading of 21.2 → 108.0
totals combine by addition
new mean = 108.08 = 13.5
divide by the new count, 8, not the old 7
The “what score do I need” item is the same move run backwards. Six quiz scores average 81; what does a seventh have to be for the average of all seven to reach 83?
target sum = 7 × 83 = 581
what the seven scores must add to
current sum = 6 × 81 = 486
what the six already add to
seventh score = 581 − 486 = 95
the gap between the two totals
The average of two averages is not the average
A conference has 45 engineers with a mean age of 38 and 30 students with a mean age of 22. The mean age of all 75 people is not38 + 222 = 30. Go through the totals: 45(38) + 30(22) = 1,710 + 660 = 2,370, and 2,37075 = 31.6. The bigger group pulls the answer toward its own mean, and 30 is always among the choices.
4
Read the value off the display, then apply the rate to it.
Half of these items are one multiplication guarding a graph you have to read correctly first.
The utility charges $0.146 per kilowatt-hour.
greatest month: April, 1,250 kWh
read the bar, not the month you expected
0.146 × 1,250 = $182.50
rate × quantity, units dollars
“how much more than March?” → 0.146(1,250 − 860)
subtract inside, multiply once — not two bills
0.146 × 390 = $56.94
same answer as 182.50 − 125.56, half the keystrokes
Keep the units attached
kWh × dollars/kWh leaves dollars, so the multiplication is right way up. If the stem had given “kilowatt-hours per dollar” instead, you would divide. Writing the units beside the numbers catches an upside-down rate faster than re-reading the sentence does.
5
Two tables: match the rows, multiply across, then add down.
One table holds counts, the other holds a rate for each category. Nothing is hard; the work is keeping the four products straight.
Item
Units sold
Price each
Revenue
Notebooks
145
$3.25
$471.25
Pens
320
$1.40
$448.00
Binders
88
$6.75
$594.00
Backpacks
41
$24.50
$1,004.50
Total
594
—
$2,517.75
145(3.25) + 320(1.40) + 88(6.75) + 41(24.50)
the whole expression, written before computing
471.25 + 448 + 594 + 1,004.50 = $2,517.75
one line in the calculator, no transcribing
largest revenue: backpacks, $1,004.50
from the fewest units of all — 41
backpack share: 1,004.502,517.75 = 39.9%
the total you just built is the denominator
The biggest count is not the biggest total
Pens outsold backpacks almost eight to one and brought in less than half as much. Any question about money has to travel through price × quantity, and any question about how many items sold stays in the count column. A choice built from the wrong column is always on offer.
6
Use Desmos as the scratch pad, not as four separate sums.
Typing the expression in one go removes the step where you copy 471.25 down and lose a digit.
type 145(3.25)+320(1.40)+88(6.75)+41(24.50)
implied multiplication works; the answer updates live
stack a fraction with the / key
61198 stacks itself, so the denominator stays visible
keep the parentheses in 0.146(1250 − 860)
without them the subtraction happens after the rate
Before you select a choice, check what was asked for one more time: a percent or a count, dollars or cents, a mean or a total, this group or all of them. On these items the arithmetic is rarely the thing that goes wrong — the answer to a slightly different question is.
✓
Check: the same four numbers, four different answers.
percent of a subtotal61198 = 30.8%the column the sentence named ✓
weighted mean2,37075 = 31.6totals, never the mean of means ✓
rate × reading0.146(1,250) = $182.50units cancel to dollars ✓
two tables41(24.50) = $1,004.50fewest units, most money ✓
Remember the flow
What is asked · which numbers · one expression · compute once · round last
The calculator is allowed. The bookkeeping is still yours.
Now you try
Practice problem
To the nearest dollar, what is the total amount paid?
A caterer buys 14 trays of sandwiches at $38.50 each and 9 trays of pastries at $26.75 each, then pays a service charge of 8.5% of the cost of the trays.
Write it all first: 14(38.50) + 9(26.75) = 539 + 240.75 = $779.75 for the trays, and the service charge adds 8.5%, so the total is 779.75 × 1.085 = 846.03 → $846. B is the tray cost alone, $779.75, with the service charge never applied — the most common miss, because the last clause is easy to read past. A subtracts the charge instead of adding it, 779.75 − 66.29 = $713.46. D moves the decimal point, treating 8.5% as 85%: 779.75 × 1.85 = $1,442.54.
Hint: two products, one sum, one percent increase — write all four pieces on one line before you press anything, and check at the end whether the question wanted the charge or the total.
