Logical Reasoning · Arrangements & Puzzles
Arithmetic Reasoning — Questions & Answers
Arithmetic reasoning questions are word problems solved with simple arithmetic and careful reading — ages, counts, legs and heads, money. Turn each sentence into a quantity.
- 5 solved questions
- Step-by-step AI walkthroughs
- How-to-approach tips
How to approach
Assume everything is the kind with fewer legs, count the "extra" legs, and divide by the difference per animal. For equal counts of different notes, add one of each and divide.
Count by groups: digits by page range, games by players knocked out, pairs as n(n − 1)/2, routes by multiplying, posts as gaps + 1. For back-and-forth counting, find the repeat length first.
Write each condition as an equation and remember that a person is not their own brother or sister. For "cannot be" questions, find what the total must be a multiple of.
Imagine the unluckiest possible draws, then add one. That extra draw forces the result.
5 solved Arithmetic Reasoning questions
A farm has hens and goats. Together they have 50 heads and 140 legs. How many goats are there?
Solution — answer C
- If all 50 were hens there would be 100 legs; the extra 40 legs come 2 per goat, so there are 20 goats.
5 steps · reveal one at a timeStep 1: Identify the Animals
We have two types of animals: hens and goats. Each animal has 1 head. Hens have 2 legs, and goats have 4 legs. We know there are 50 heads and 140 legs in total.
Step 2: Assume All Hens
Let's imagine for a moment that all 50 animals were hens. If this were the case, the total number of legs would be 50×2=100 legs.
Step 3: Find the Extra Legs
However, the problem states there are 140 legs. The difference between the actual legs and our 'all hens' assumption is 140−100=40 legs. These are the extra legs.
Step 4: Calculate Goats
Each goat has 4 legs, while each hen has 2 legs. So, replacing a hen with a goat adds 4−2=2 extra legs. To account for the 40 extra legs, we divide 40÷2=20. There are 20 goats.
Step 5: The answer
So the answer is (C) 20. There are 20 goats, as each goat contributes 2 extra legs compared to a hen, making up the difference from an all-hen scenario.
How many digits are used in numbering the pages of a book from 1 to 250?
Solution — answer D
- Pages 1–9 use 9 digits, 10–99 use 90 × 2 = 180, and 100–250 use 151 × 3 = 453: 9 + 180 + 453 = 642.
4 steps · reveal one at a timeStep 1: Count 1-digit pages
First, let's count the digits for pages that only use one digit. These are pages 1 through 9. There are 9 such pages, so they use 9×1=9 digits in total.
Step 2: Count 2-digit pages
Next, consider pages that use two digits. These range from page 10 to page 99. To find how many pages there are, calculate 99−10+1=90 pages. Each uses two digits, so this section uses 90×2=180 digits.
Step 3: Count 3-digit pages
Finally, we have pages with three digits, starting from page 100 up to the last page, 250. The number of pages is 250−100+1=151 pages. Each page uses three digits, so this section uses 151×3=453 digits.
Step 4: The answer
So the answer is (D) 642. To find the total number of digits, we add up the digits from each section: 9+180+453=642 digits.
In a family, each boy has as many sisters as brothers, and each girl has twice as many brothers as sisters. How many girls are there?
Solution — answer B
- With b boys and g girls: b − 1 = g and b = 2(g − 1). So g + 1 = 2g − 2, giving g = 3 (and 4 boys).
5 steps · reveal one at a timeStep 1: Define variables
Let 'b' be the number of boys and 'g' be the number of girls in the family. We'll use these to set up equations based on the given conditions.
Step 2: First condition
Each boy has as many sisters as brothers. A boy has 'g' sisters and 'b−1' brothers. So, we can write the equation: g=b−1.
Step 3: Second condition
Each girl has twice as many brothers as sisters. A girl has 'b' brothers and 'g−1' sisters. This gives us the second equation: b=2(g−1).
Step 4: Solve the equations
Now we have two equations: g=b−1 and b=2(g−1). From the first, we know b=g+1. Substitute this into the second equation: g+1=2(g−1). This simplifies to g+1=2g−2, which gives g=3.
Step 5: The answer
So the answer is (B) 3. There are 3 girls in the family, and substituting back, there are 3+1=4 boys, which satisfies both conditions.
A drawer has 10 red and 10 blue socks, all loose. In the dark, how many socks must you take to be sure of getting two blue socks?
Solution — answer A
- In the worst case the first 10 are all red; two more are then certainly blue, so 12.
5 steps · reveal one at a timeStep 1: Understand the Goal
We need to find the minimum number of socks to pick to guarantee having two blue socks. This means considering the worst possible scenario to ensure success.
Step 2: Worst Case Scenario
To be absolutely sure of getting two blue socks, we must consider the unluckiest sequence of picks. This involves picking all the socks that are not blue first.
Step 3: Pick All Red Socks
In the worst case, you would first pick all the red socks available. There are 10 red socks in the drawer, so you pick these 10 socks first.
Step 4: Guarantee Blue Socks
After picking 10 red socks, only blue socks are left in the drawer. The next two socks you pick must be blue. So, you need to pick 2 more socks.
Step 5: The answer
So the answer is (A) 12. You pick 10 red socks, then 2 more blue socks, for a total of 10+2=12 socks to guarantee two blue.
If Ravi gives Siva 5 marbles, they will have the same number. If Siva gives Ravi 5 marbles, Ravi will have twice as many as Siva. How many marbles does Ravi have?
Solution — answer A
- The first condition gives R = S + 10; the second gives R + 5 = 2(S − 5). So S + 15 = 2S − 10: S = 25 and R = 35.
5 steps · reveal one at a timeStep 1: Define variables
Let's assign variables to the number of marbles each person has. Let Ravi have R marbles and Siva have S marbles. This helps us translate the problem into mathematical equations.
Step 2: First condition equation
The first condition states: "If Ravi gives Siva 5 marbles, they will have the same number." This means R−5=S+5. We can simplify this to R=S+10. This is our first equation.
Step 3: Second condition equation
The second condition states: "If Siva gives Ravi 5 marbles, Ravi will have twice as many as Siva." This translates to R+5=2(S−5). This is our second equation.
Step 4: Solve the equations
Now we substitute the first equation (R=S+10) into the second one: (S+10)+5=2(S−5). This simplifies to S+15=2S−10. Solving for S, we get S=25. Then, using R=S+10, we find R=25+10=35.
Step 5: The answer
So the answer is (A) 35. Ravi has 35 marbles, which satisfies both conditions when checked.
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