Digital SAT Prep · Digital SAT Reading, Writing & Math

Problem-Solving and Data Analysis

Ratios, rates, percentages, and one and two variable data interpretation.

Four ideas cover Digital SAT Problem-Solving and Data Analysis: percent of a quantity, percent change (and successive factors), ratios and rates with unit conversion, and reading a small data set for mean, median and range. Every trap in this domain is a wrong base, a wrong factor, or a mean that followed an outlier the median ignored.

Digital SAT Prep
Medium level
6 concepts
5 practice questions

1A percent of a quantity is multiplication by a decimal

x% of a whole means (x/100) times that whole. Convert the percent to a decimal or a fraction before multiplying — 35% of 240 is 0.35 × 240, not a proportion you have to solve for. The same conversion runs in reverse: if 40% of a number is 56, the number is 56 ÷ 0.40.

"Percent of" statements that look like equations are usually just this product written in words. Set up a variable only when the percent is of an unknown that also appears elsewhere; otherwise multiply.

Figure. 35% of 240 is the 84 segment; the whole bar is the quantity the percent acts on.

How it works

Name the wholeThe number after "of" is the whole; the percent acts on it.
Convert, then multiplyWrite x% as x/100 (or a known fraction) and multiply by the whole.
Reverse when given the partIf you know the part and the percent, divide the part by the decimal form.

Part from a percent, then reverse

Find 35% of 240. Then find the number of which 40% equals 56.

35% = 0.35, so 0.35 × 24084
40% = 0.40, and part = 56part = 0.40 × whole
whole = 56 ÷ 0.40140
Check: 0.40 × 14056

Pro tip. Dividing by the decimal is the whole skill when the part is given and the whole is asked — multiplying instead gives 40% of 56, which is the trap option on almost every reverse-percent item.

15% of a number is 45. The number is

45 ÷ 0.15 = 300. Multiplying 45 × 0.15 gives 6.75 (near the distractor shape of "3" if the decimal drifts), and 45 × 15 = 675 treats the percent as a whole number instead of a hundredths factor.

2Percent change divides by the original value

Percent change equals (new − old) / old × 100. Increases come out positive and decreases negative, but the sign is secondary — the trap is the denominator. It is always the value you are changing from, never the new value and never whichever number is larger.

That choice makes the relationship asymmetric. A rise from 40 to 46 is a 15% increase, but the fall from 46 back to 40 is only about 13% — the same difference of 6 measured against a bigger base.

Figure. The rise is 6; percent change puts that 6 over the old bar of 40, not over 46.

How it works

Name the originalUnderline the starting value — that number alone sits under the fraction.
Take the differenceNew minus old; keep the sign so a fall stays negative.
Divide by original, × 100Never divide by the new value when the question asks for percent change.

Percent increase on a price

A book's price rose from $40 to $46. What was the percent increase?

Change = 46 − 406
Base is the original, 406 / 40
6 / 400.15
0.15 × 10015%

Pro tip. Dividing by the new price (6/46) gives about 13% — that is the percent decrease from 46 back to 40, not the increase that happened. Always divide the change by the ORIGINAL value.

A quantity falls from 50 to 40. The percent decrease is

(50 − 40)/50 × 100 = 20%. Using 40 as the base gives 25%, which is the percent increase from 40 back up to 50 — the mirror-image trap of switching the base.

3Successive percent changes multiply as factors

Two percent changes in a row are not added. A rise of 20% followed by a fall of 10% multiplies by 1.20 and then by 0.90, which is 1.08 — an 8% net rise, not a 10% rise. Adding the signed percents ignores that the second change acts on a different base.

The compact formula a + b + ab/100 (with signs) recovers the same net percent: 20 + (−10) + (20)(−10)/100 = 8. Equal-and-opposite successive changes never cancel, because the second percentage is charged on the already-changed amount.

Figure. Start 100, then 120, then 108 — net +8 on the original, not +10.

How it works

Write each change as a factor+x% is ×(1 + x/100); −x% is ×(1 − x/100).
Multiply the factorsThe product is the single factor from start to finish.
Convert back to a percentNet percent = (product − 1) × 100, or use a + b + ab/100 with signs.

Plus 20%, then minus 10%

A price rises by 20% and then falls by 10%. What is the net percent change from the original price?

After +20%: factor1.20
After −10%: factor0.90
1.20 × 0.901.08
Net change = (1.08 − 1) × 1008% increase

Pro tip. On a starting value of 100 the path is 100 → 120 → 108, so the net is visibly +8, not +10. The same arithmetic on successive discounts of 20% and 15% gives 0.80 × 0.85 = 0.68 — a 32% total reduction, not 35%.

Successive discounts of 20% and 15% equal a single discount of

0.80 × 0.85 = 0.68, so 32% comes off. Adding 20 and 15 gives 35% and always overstates, because the second discount acts on a price the first has already reduced.

4Ratios scale; proportions cross-multiply

A ratio a : b can be scaled by any common factor — 3 : 5 is the same comparison as 6 : 10 or 12 : 20. When the parts must add to a known total, write a = 3x and b = 5x so that 8x equals the total, then solve for x.

A proportion says two ratios are equal: a/b = c/d. Cross-multiplication gives ad = bc and is the fastest clean algebra for a missing term. Check that you paired corresponding quantities before crossing.

Figure. Eight equal parts make 40, so each part is 5 — three parts 15, five parts 25.

How it works

Scale with a common factorIf only the ratio is known, multiply both parts by the same x.
Fit a totalWhen parts add to T, set (sum of ratio numbers) × x = T.
Cross-multiply a proportionFrom a/b = c/d write ad = bc and solve the one unknown.

Split a total in a given ratio

Split 40 in the ratio 3 : 5. Then solve 4/x = 6/15 for x.

Parts 3x and 5x; 3x + 5x = 408x = 40
x = 40 / 85
Parts: 3×5 and 5×515 and 25
4/x = 6/15 ⇒ 4 × 15 = 6xx = 10

Pro tip. The cross-multiply row is one product on each side — writing 4 × 6 = 15x is the classic pairing error that swaps which terms correspond.

If 3 : 5 = 12 : x, then x is

3/5 = 12/x ⇒ 3x = 60 ⇒ x = 20. Answering 15 scales only one side, and 9 comes from subtracting rather than cross-multiplying.

5Rate is quantity over time; convert units with factors

A rate is quantity divided by time — miles per hour, litres per minute, words per minute. Once the rate is known, distance (or amount) is rate × time, and time is quantity ÷ rate. Keep the units attached so a cancelled unit in a conversion chain is visible.

Unit conversion is multiplication by factors that equal 1: 1000 m / 1 km, 3600 s / 1 h. Chain them so unwanted units cancel and the units you want remain. Mixing km/h with minutes without converting is the usual error, not the arithmetic.

Figure. Distance against time at 60 mi/h is a straight line through the origin; at 2.5 h the height is 150 miles.

How it works

Form the raterate = quantity / time in matching units.
Apply rate × timeOr rearrange: time = quantity / rate.
Insert conversion factorsMultiply by fractions equal to 1 until only the target units remain.

Constant speed, then a unit change

A car covers 180 miles in 3 hours at constant speed. How far does it travel in 2.5 hours at that speed? Also convert 72 km/h to m/s.

rate = 180 / 360 mi/h
distance in 2.5 h = 60 × 2.5150 miles
72 km/h = 72 × (1000 m)/(1 km) × (1 h)/(3600 s)72 × 1000 / 3600
72 × 5/1820 m/s

Pro tip. The km/h → m/s shortcut multiply-by-5/18 is just the conversion-factor row collapsed; if you forget which way the 5/18 goes, write the units out once and you will see it.

At 90 km/h, the time to cover 60 km is

40 minutes
1.5 hours
2/3 minutes

time = 60/90 = 2/3 hour = 40 minutes. Leaving the answer as 2/3 without converting units mismatches the usual minute options; 1.5 hours inverts the fraction.

6Mean follows the total; median resists outliers

The mean is the sum divided by the count — so recover a sum as mean × count before adding or removing a value. The median is the middle value once the list is ordered. The range is max minus min.

A few large outliers pull the mean toward them and barely move the median. When a data set gains an extreme value, expect the mean to shift and the median to stay put unless the new value changes which entry sits in the middle.

Figure. Values 2, 4, 6, 8, 20: the mean sits at 8, pulled up by 20; the median stays at the middle value 6.

How it works

Recover the sumsum = mean × count before any add/remove.
Adjust the sum, then divideAdd the new value (or subtract a removed one) and divide by the new count.
Order for median and rangeSort once; median is the middle (or mean of two middles); range is max − min.

Mean after adding a value

Five numbers have a mean of 12. A sixth number, 30, is added. What is the new mean?

Sum of the first five = 5 × 1260
New sum = 60 + 3090
New count6
New mean = 90 / 615

Pro tip. Recover the total from mean × count; adding or removing values is always easiest via the sum. On the list 2, 4, 6, 8, 20 the mean is 8 and the median is 6 — remove the outlier 20 and the mean falls to 5 while the median only moves to 5.

The mean of five scores is 84. A sixth score of 90 is included. The new mean is

Old sum 420; new sum 510; 510/6 = 85. Averaging 84 and 90 gives 87 and pretends the new score replaces the whole set rather than joining it.

Notes

Percent Change: Percent change equals the difference divided by the original value, times 100; increases are positive and decreases negative.
Ratios and Proportions: A ratio a:b can be scaled by any common factor; solve proportions by cross-multiplication \frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc.
Rates and Unit Conversion: Use \text{rate}=\frac{\text{quantity}}{\text{time}} and multiply by conversion factors to change units consistently.
Statistics from Data: The mean is the average, the median is the middle value, and the range is max minus min; a few large outliers pull the mean but not the median.

Formulas

Percent change: \dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%
Proportion: \dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc
Mean: \bar{x} = \dfrac{\text{sum of values}}{\text{number of values}}
Percent of a number: \text{part} = \dfrac{\text{percent}}{100}\times \text{whole}
Unit rate: \text{rate} = \dfrac{\text{quantity}}{\text{time}}

Exam traps & shortcuts

For successive percent changes, multiply factors (e.g., +20% then -10% is \times1.2\times0.9=1.08), never add the percentages.
When a dataset gains an outlier, expect the mean to shift toward it while the median barely moves.
Convert 'percent of' statements into multiplication by a decimal to avoid setting up unnecessary equations.

Reference tables

Every row should be reconstructible from the concept that owns it, not merely recalled the night before.

Formula sheet
Quantity	Relation	Watch for
Percent of a number	part = (percent/100) × whole	Convert to a decimal before multiplying
Percent change	(new − old)/old × 100%	Base is always the original value
After a change	new = old × (1 ± x/100)	Chain factors for successive changes
Successive changes	a + b + ab/100 (signed)	Never add the percents alone
Proportion	a/b = c/d ⇒ ad = bc	Pair corresponding quantities
Unit rate	rate = quantity / time	Convert units before dividing
Mean	sum / count	Recover sum as mean × count first
Median / range	middle of ordered list; max − min	Outliers move the mean, not the median

Recap

Read only this the night before.

Percent of: Multiply by the decimal form. Given the part, divide by that decimal to recover the whole.
The base: Percent change divides by the value you came from. The same difference against a new base is a different percent.
Successive: Multiply factors, or use a + b + ab/100. Equal up-then-down never returns you to the start.
Ratio: Scale with x; fit a total with (sum of parts)×x. Cross-multiply a proportion.
Rate: quantity/time, then × time. Convert units with factors of 1 so cancelled units show.
Data: Mean via the sum; median from the ordered middle; an outlier pulls the mean and barely moves the median.

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