- Use the diagram to answer this question. Alex drew circles around groups of stars he counted in the sky. Circle X has some stars and Circle Y has some stars. Alex's little sister then draws 1 more star in Circle Y. After she draws the extra star, Circle Y has more stars than Circle X. What must have been true about the circles before the sister drew the extra star?
Answer: Circle X and Circle Y had exactly the same number of stars before adding extra
If adding 1 star to Circle Y makes Circle Y have more stars than Circle X, and we see from the diagram that Circle X has 4 stars and Circle Y has 3 stars, then originally they had 4 and 3 respectively. But for the logic to work where adding 1 to Y makes Y have more than X, X and Y must have started with the same number. After adding 1 to Y, Y would have 1 more than X.
- Maya counts her stickers in two boxes. Box A has 7 stickers and Box B has 4 stickers. Then Maya's friend gives her 2 more stickers to put in Box B. Now compare the number of stickers in Box A to the number of stickers in Box B.
Answer: Box A has more stickers than Box B after Maya gets the new stickers
Originally Box A had 7 stickers and Box B had 4 stickers. After Maya gets 2 more stickers for Box B, Box B has 4 + 2 = 6 stickers. Since 7 > 6, Box A still has more stickers than Box B.
- Look at the picture. Emma has some red blocks and some blue blocks. After she gives away 2 red blocks, she counts and finds she has the same number of red blocks as blue blocks. What can you say about how many blocks Emma had at the beginning?
Answer: Emma had more red blocks than blue blocks at the beginning
If Emma had the same number of red and blue blocks after giving away 2 red blocks, then she must have started with 2 more red blocks than blue blocks. This means she had more red blocks than blue blocks at the beginning. Choice B is incorrect because having fewer would mean she'd have even fewer after giving some away. Choice C is incorrect because if she started with equal amounts and gave away 2 red, she'd have fewer red than blue. Choice D is backwards - she had 2 more red blocks, not 2 fewer.
- Look at the picture showing two groups of flowers. Group 1 has some sunflowers and Group 2 has some roses. Tom picks 2 sunflowers from Group 1. After he picks them, Group 1 and Group 2 have the same number of flowers. How did the groups compare before Tom picked any flowers?
Answer: Group 1 had exactly 2 more flowers than Group 2 before Tom picked any
If Group 1 and Group 2 have the same number of flowers after Tom picks 2 flowers from Group 1, then Group 1 must have started with exactly 2 more flowers than Group 2. When Tom removed 2 flowers from Group 1, it made the groups equal. Choice B is incorrect because if Group 2 had more originally, Group 2 would have more after Tom picked from Group 1. Choice C is incorrect because if they started equal, Group 1 would have fewer after picking 2. Choice D is the same as Choice B and is also incorrect.
- Sarah has two boxes of crayons. The green box starts with 8 crayons. The yellow box starts with 5 crayons. Sarah's mom puts 2 more crayons in the yellow box and takes 1 crayon out of the green box. After these changes, compare the number of crayons in each box.
Answer: The green box and yellow box have the same number of crayons after changes
Originally: Green box = 8 crayons, Yellow box = 5 crayons. After changes: Green box = 8 - 1 = 7 crayons, Yellow box = 5 + 2 = 7 crayons. Since both boxes have 7 crayons, they have the same number. Choice A is incorrect because 7 is not greater than 7. Choice B is incorrect because 7 is not greater than 7. Choice D is incorrect because 7 is not twice 7.
- Lily has a jar of marbles. She takes out 5 marbles and counts them: 2 are red and 3 are blue. Then she takes out 3 more marbles and they are all red. Now compare the total number of red marbles to the total number of blue marbles that Lily has taken out.
Answer: Lily has taken out more red marbles than blue marbles from the jar
When you see a problem about comparing groups of objects, you need to carefully count each type and then compare the totals. Let's track Lily's marbles step by step.
First, Lily takes out 5 marbles: 2 red and 3 blue. Then she takes out 3 more marbles that are all red. To find the totals, you add up each color separately:
Total red marbles = 2 (from first group) + 3 (from second group) = 5 red marbles
Total blue marbles = 3 (from first group) + 0 (from second group) = 3 blue marbles
Now compare: 5 red marbles is more than 3 blue marbles, so Lily has taken out more red marbles than blue marbles.
Looking at the wrong answers: Choice A says more blue than red, but we found 3 blue versus 5 red, so this is backwards. Choice C claims the same number of each color, but 5 red ≠ 3 blue. Choice D states exactly twice as many blue as red, which would mean blue = 2 × red, but we have 3 blue and 5 red, and 3 ≠ 2 × 5.
The correct answer is B because 5 red marbles is greater than 3 blue marbles.
For counting problems like this, always organize your work by making separate tallies for each category, then carefully add up each group before comparing. This prevents you from mixing up the numbers or forgetting to include all the items from each round of counting.
- Sam has some toy cars in a red box and some toy cars in a blue box. He counts and finds that the red box has 3 more cars than the blue box. If Sam moves 1 car from the red box to the blue box, what will be true about the number of cars in each box?
Answer: The red box will have more cars than the blue box after moving one car
When you see a problem about moving objects between groups, you need to think about how the movement affects both groups at the same time.
Let's start with what we know: the red box has 3 more cars than the blue box. You can picture this with any numbers that fit - like if the blue box has 2 cars, then the red box has 5 cars.
Now when Sam moves 1 car from red to blue, two things happen: the red box loses 1 car, and the blue box gains 1 car. So the red box goes from 5 cars to 4 cars, and the blue box goes from 2 cars to 3 cars. The red box still has more cars (4 is more than 3), which makes D correct.
Let's check why the other answers are wrong. Choice A says the red box will still have exactly 3 more cars, but that's not true - it now has only 1 more car than the blue box (4 minus 3 equals 1, not 3). Choice B claims the blue box will have more cars, but 3 cars is still less than 4 cars. Choice C suggests they'll be equal, but 4 and 3 are not the same number.
Here's your key strategy: when objects move between groups, the difference between the groups changes by 2. Moving 1 item reduces the original difference by 2 (from 3 more to 1 more in this case). Remember that the group that loses items goes down while the group that gains items goes up.
- Katie is organizing her toy collection. She puts her dolls in one basket and her stuffed animals in another basket. The basket with dolls has 6 toys. The basket with stuffed animals has some toys, but Katie knows it has fewer toys than the doll basket. What is a possible number of stuffed animals in the second basket?
Answer: The stuffed animal basket could have exactly 4 stuffed animals in it
The doll basket has 6 toys, and the stuffed animal basket has fewer toys than the doll basket. This means the stuffed animal basket must have less than 6 toys. Only choice C (4 stuffed animals) is less than 6. Choice A is incorrect because 8 > 6. Choice B is incorrect because 6 = 6, not fewer. Choice D is incorrect because 7 > 6.
