GRE · Arithmetic · TOPIC LESSON
Choose the reference quantity, align units and model rates before calculating.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
If red:blue=3:5, write red=3k and blue=5k with k>0 for positive amounts. The total is 8k, so red is 3/8 of the total, not 3/5. If counts are integers, the scale factor may face divisibility constraints; a fluid mixture need not use integer liters.
Ratios can be scaled only by the same multiplier on both terms. Adding the same amount to both groups generally changes the ratio. To combine ratios sharing a quantity, align the shared quantity before joining them.
ORIGINAL WORKED EXAMPLE
A collection has fiction:nonfiction=7:4 and 165 books in total. How many are nonfiction?
Answer 60 books
02 / UNDERSTAND & APPLY
Direct proportionality has y=kx: doubling x doubles y and y/x stays constant. Inverse proportionality has y=k/x: doubling x halves y and xy stays constant. A relationship that merely increases is not necessarily proportional; a fixed starting fee gives y=kx+b, which is not direct proportionality unless b=0.
Cross multiplication solves a/b=c/d only when b and d are nonzero. First make sure both ratios describe the same orientation and compatible units.
03 / UNDERSTAND & APPLY
Speed is distance/time, production rate is output/time, and unit price is money/item. Attach units to every conversion so unwanted units cancel. For example, 72 km/h×1000 m/km÷3600 s/h=20 m/s. A correct number with the wrong unit is not a correct answer.
For a constant rate r over time t, quantity=rt. If rates vary, split the journey or task into segments and add quantities or times as appropriate. Average speed is total distance divided by total time, not usually the average of speeds.
ORIGINAL WORKED EXAMPLE
A cyclist travels 12 km at 18 km/h and returns 12 km at 12 km/h. Find the average speed for the trip.
Answer 14.4 km/h
04 / UNDERSTAND & APPLY
If one machine finishes a job in a hours, its constant rate is 1/a job per hour. Two independent machines working simultaneously have rate 1/a+1/b. Completion time is the reciprocal of that combined rate. Times themselves do not add.
A drain opposes a filling pipe, so subtract its rate. If the net rate is zero, the amount does not grow; if negative, the tank cannot fill from empty under that model. Problems assume constant compatible rates only when stated or clearly implied.
ORIGINAL WORKED EXAMPLE
Machines complete the same job alone in 8 and 12 hours. How long together at constant rates?
Answer 4.8 hours
05 / UNDERSTAND & APPLY
p percent means p/100 of the chosen base. Percentage change is (new−old)/old×100%, provided old is nonzero; the old quantity is the reference. A percentage-point change subtracts two percentages directly and is different from their relative percentage change.
If a participation rate rises from 20% to 25%, it rises by 5 percentage points, but by 25% relative to its original rate. Saying just “up 5 percent” would be ambiguous or wrong in that context.
ORIGINAL WORKED EXAMPLE
After a 15% discount, a lamp costs $68. What was the original price?
Answer $80
06 / UNDERSTAND & APPLY
An increase of p% multiplies by 1+p/100; a decrease multiplies by 1−p/100. Successive changes multiply these factors. Equal percentage increase and decrease do not cancel because their bases differ.
A 20% rise followed by a 20% fall multiplies by 1.2·0.8=0.96, a net 4% decrease. To reverse a 20% fall, multiply by 1/0.8=1.25, a 25% increase. The order of pure percentage factors does not change the final product, though intervening fixed amounts would change that.
ORIGINAL WORKED EXAMPLE
A shop marks an item up 30% from its $50 cost, then discounts the marked price 10%. Find the sale price and profit percentage relative to cost.
Answer $58.50; 17% profit
07 / UNDERSTAND & APPLY
For a mixture, ingredient amount=concentration×total mixture amount, using compatible units and a stated interpretation of concentration. The final concentration is total ingredient divided by total mixture. It is a weighted average, so it must lie between starting concentrations when positive amounts are mixed without reaction or loss.
Combining equal volumes permits an ordinary average of concentrations; unequal volumes do not. Adding water changes total volume but does not add the original solute under the usual idealized additive-volume model. State that model rather than assuming real chemical volumes always add.
08 / UNDERSTAND & APPLY
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
09 / UNDERSTAND & APPLY
Show the model before using a calculator.
ORIGINAL PRACTICE · 1
35
The shared multiplier is 56/8=7, so A=5·7=35. Using 5/13 would find A as a share of a total, which 56 is not.
ORIGINAL PRACTICE · 2
B: 60 miles/hour
Distance/time=150/2.5=60. Multiplying distance and time gives incompatible units, not speed.
ORIGINAL PRACTICE · 3
25
Increase is $6000; divide by the original $24000 to obtain 0.25=25%. Dividing by the new value would give 20%, answering a different question.
ORIGINAL PRACTICE · 4
B
The final price is 1.1·0.9P=0.99P, less than P because P>0. Equal percentages on different bases do not cancel.
ORIGINAL PRACTICE · 5
A, B and C
Multiply by 60 for 900 m/min, divide by 1000 for 0.9 km/min, then multiply by 60 for 54 km/h. D ignores the conversion factors and E uses a factor of 10 instead of 60.
ORIGINAL PRACTICE · 6
15
Net rate=1/6−1/10=1/15 tank/hour, so time=15 hours. Adding rates would be appropriate for two filling pipes, not a drain.
ORIGINAL PRACTICE · 7
32
Solute volume=3·0.2+2·0.5=1.6 liters. Total is 5 liters, so concentration is 1.6/5=32%. The unweighted 35% average ignores unequal volumes.
ORIGINAL PRACTICE · 8
A
The values are 4 and 2. Inverse proportionality keeps xy fixed, so increasing x decreases y. A direct-proportion assumption would reverse the comparison.
KEEP THE BIG IDEAS
READ WITH CONTEXT
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