#1: 6 June 2022 :

The gray area has the same two halves as the white area and then one more half, for a total of three halves, or 3/2.

#2: 7 June 2022 :

The larger square is half of the entire figure, or 2/4 (“two fourths”). Two smaller squares are two eighths (2/8), or one-fourth (1/4), of the entire figure. Altogether, these three regions represent 2/4 + 1/4, or 3/4, of the entire figure.

#3: 8 June 2022 :

The triangle is divided into 6 regions, each of which is 1/6 of the area of the triangle. The sum of the orange region (1/6) and the pink region (2/6 “two sixths” or 1/3 “one third”) is 3/6 “three sixths” (or 1/2 “one half”) of the triangle, which is equal to the area of the green region.

#4: 9 June 2022 :

The partitions of the 2 triangles are colored into 3 groups of 2/3 (“two thirds”), so each 2/3 is 1/3 of 2. In other words, each colored area is 1/3 of the total area of 2 triangles.

#5: 10 June 2022 :

If one kid gets 3/4 (“three fourths”) of one waffle, then 4 kids need 4 x 3/4, or 12/4 (“twelve fourths”), waffles. Twelve fourths of a waffle is equal to 3 waffles.

#6: 11 June 2022 :

There are 4 blobs in 2/3 of an order. So there are 2 blobs in 1/3 of an order. That means there are 6 blobs in one order. Half of one order of 6 blobs is 3 blobs.

#7: 12 June 2022 :

The red square is 1/6 of the entire rectangle. The shaded region is 1/3 of that red square. So the shaded region is also 1/3 of 1/6, or 1/18, of the entire rectangle. In other words, 18 of those shaded region fit into the entire rectangle.

#8: 13 June 2022 :

1/4 is equivalent to 2/8, and 2/8 is at the second tick mark from 0. That means that the tick marks are 1/8 apart. And that means that 9/8 is at the 9th tick mark from 0, which is Point F.

#9: 14 June 2022 :

1/3 of the triangle is 1 region, so 2/3 of the triangle is 2 regions. We need 3/2 (“three halves”) of those 2 regions. 1/2 (“one half”) of 2 regions is 1 region; 2/2 (“two halves”) of 2 regions is 2 regions; and 3/2 (“three halves”) of 2 regions is 3 regions, which is equal to the whole triangle. This means that 3/2 of 2/3 of the triangle is equal to 1 triangle.

#10: 15 June 2022 :

1/3 more than 1 bunch is 1 + 1/3, or 4/3, of a bunch of daisies. There are 8 daisies in 4/3 of a bunch, which means that there are 2 daisies in 1/3 of a bunch. That means there are 6 daisies in 1 bunch.

#11: 16 June 2022 :

When Kim took 1/6 of the 2 cakes, she took 2 pieces. When Paolo took 1/5 of the 10 pieces that remained, he also took 2 pieces. So Kim and Paolo got the same-size share. The reason we can’t compare 1/6 and 1/5 to find the answer is because the share of the cake that Kim and Paolo took from were different sizes.

#12: 17 June 2022 :

## The leaf is 1/3 + 2 + 1/6 inches, or 1/6 + 1/6 + 2 + 1/6 inches, which is equal to2 and 1/2 inches.

#13: 18 June 2022 :

4 waffles ÷ 2 = 2 waffles, 2 waffles × 3 = 6 waffles. 4 ÷ 2 × 3 = 4 × (1/2) × (3/1) 4 waffles × (3/2) = 6 waffles.

#14: 19 June 2022 :

The orange section of the rectangle contains 16 oranges, so each 1/4th of the orange section contains 4 oranges. The banana section contains 6 of these 1/4-size regions, so there are 6 × 4 = 24 bananas in the banana section. The apple section contains 2 of these 1/4-size regions, so there are 2 × 4 = 8 apples in the apple section.

#15: 20 June 2022 :

Remove the same glasses from each set. That leaves us with 2 small glasses in Set A and 1 large glass in Set B. Since both sets hold the same amount of liquid, then the two small glasses in Set A hold the same amout of liquid as a large glass in Set B. Therefore, a small glass contains 1/2 the volume of a large glass.

#16: 21 June 2022 :

The 3 blue sections are 3/4 of the square; the 5 blue squares are 5/36 of the square; and the 4 pink squares are 1/9 of the square. So the equation is 3/4 + 5/36 + 1/9 = 1. A=3, B=5, and C = 1.

#17: 22 June 2022 :

There are many ways to do this. I folded the paper in 1/2, then in 1/2 again, then in 1/2 again, and then in 1/3s. This divides the paper into 2 × 2 × 2 × 3 = 24 regions. Another way of thinking about this is that after all the folding, each region is 1/2 × 1/2 × 1/2 × 1/3 = 1/24th of the size of the original paper.

#18: 23 June 2022 :

As shown in the picture, 1 brick weighs the same as 1 pound + 1/2 a brick. That means that 1/2 a brick weighs 1 pound. [Cover up the bricks to the right of the dotted line to see it.] And that means that 1 brick weighs 2 pounds.

#19: 24 June 2022 :

In 2 and 1/2 of the 3 hexagons, I shaded 5/6s (“five sixths”) in blue, then another 5/6s in green, and then another 5/6s in pink. That means there are three 5/6s in 2 1/2.

#20: 25 June 2022 :

The three shaded bars are 3/5 of the entire rectangle. 1/3 of those three bars is one bar; 2/3 are two bars; 3/3 are three bars; 4/3 are four bars; and 5/3 are five bars. Five bars is the entire rectangle. This shows that 5/3 of 3/5, or 5/3 × 3/5, is equal to 1.

#21: 26 June 2022 :

1/3 of the candy bar is 4 pieces, and 1/2 of the candy bar is 6 pieces. So the question is, how many times will 4 go into 6? 1 1/2 times. This means that 1/2 ÷ 1/3 = 1 1/2.

#22: 27 June 2022 :

Since 2/3 of the bookshelf is filled by 3/5 of the book collection, then 1/3 of the bookshelf is filled by 1/2 of 3/5, or 3/10, of the book collection. And therefore, 3 × 1/3, or 3/3 (the entire bookshelf), of the bookshelf is filled by 3 × 3/10, or 9/10, of the collection. 9/10 of the book collection will fill the entire bookshelf.

#23: 28 June 2022 :

Here are two solutions. In the first solution, the sequence of blue arrows shows the sum (in order) of 1/3 + 5/12 + 1/4. In the second solution, the sequence of blue arrows shows the sum (in order) of 2/3 + 1/12 + 1/4.

#24: 29 June 2022 :

• 9/10 is 1/10 from 1.

• 10/9 is 1/9 from 1.

• 1/9 is greater than 1/10 (i.e., 1/9 > 1/10), so 10/9 is farther from 1 than 9/10 is. You can see this in the picture.

• Which means that 9/10 is closer to 1 than 10/9 is.

#25: 30 June 2022 :

My strategy was somewhat trial-and-error here. To find the smallest positive difference, I found the two fractions that were closest together. To find the largest positive difference, I first found the largest possible fraction and then I created the smallest fraction from the digits that remained.

#26: 1 July 2022 :

The question I want to answer is, how many halves (the brown piece) fit into a sixth (the olive piece)? Well, as shown in the second figure, 1 half contains 3 sixths. Also, 1/3 of a brown piece (1/2) is the same as an olive piece (1/6). Therefore, 1/3 of a half fits into a sixth: 1/6 ÷ 1/2 = 1/3.

#27: 2 July 2022 :

1. The area of a green square is 1/4 of the area of the entire figure.

2. The 9 blue squares that fill up the area of a green square represent 9/36 of the area of the entire figure.

3. The 2 1/4 (“2 and 1/4”) red squares that fill up the area of a green square represent 2 1/4 / 9 of the area of the entire figure.

## Did you notice that the three fractions are equivalent: 1/4 = 9/36 = 2 1/4 / 9. Why is that? Think about it and then click here and I'll tell ya.

The three fractions are equivalent, because they all describe the same area (the green square) as a fraction of the area of the entire figure.

#28: 3 July 2022 :

For n = 1, the pink region is 1/3 of the entire triangle. For n = 2, the sum of 2 pink regions is 2/6, or 1/3 of the area of 2 triangles. For n = 3, the sum of 3 pink regions is 3/9, or 1/3, of the area of 3 triangles. Generalizing from these three cases, the answer is 1/3. This is because for any n, there are n pink regions out of 3n triangles: n/3n = 1/3.

#29: 4 July 2022 :

The daisies in the blue rectangle are 1/3 of the daisies in the green rectangle, which are 2/3 of the daisies in the orange rectangle, which are 3/4 of the daisies in the entire arrangement of daisies.

Also, the daisies in the blue rectangle are 1/3 of 2/3 of 3/4 = 1/3 × 2/3 × 3/4 = (1 × 2 × 3)/(3 × 3 × 4) = 1/6 of the daisies in the entire arrangement of daisies.

#30: 5 July 2022 :

As shown in the redrawn figure, the height of the red rectangle is 1/3 of the 12 cm height of the square, or 4 cm. The width of the red rectangle is 1/2 + 1/6 = 2/3 of the 12 cm width of the square, or 8 cm. The red rectangle has dimensions 4 cm × 8 cm, so its area is 32 square centimeters.

#31: 6 July 2022 :

My answers are given in the annotated figure.

1) 3 + 4 + 5 = 12

2) Each fraction in the equation is the name of one of the three regions in the polygon. The sum of those fractions is 1, which represents the entire polygon.

#32: 7 July 2022 :

The new prism has dimensions 2 × 3 × 4, so its volume is 24.

1. The volume of the new prism is 24/60, or 2/5, of the volume of the original one.

2. 2/3 × 3/4 × 4/5 = (2 × 3 × 4)/(3 × 4 × 5) = 2/5

#33: 8 July 2022 :

The center square at the bottom of the figure is 1/3 of 1/2, or 1/6, of the rectangle. We're looking for 2/9 of the figure and that center square covers 1/6 of it. So we still need to find 2/9 – 1/6 = (2/9 × 2/2) – (1/6 × 3/3) = 4/18 –3/18 = 1/18 more.

The light blue square is also 1/6 of the entire figure, and three small squares within it are 1/3 of the blue square. So three small squares are 1/3 of 1/6, or 1/18, of the rectangle, and that's the additional 1/18 we need.

Therefore, the regions shaded yellow make up 2/9 of the rectangle.

#34: 9 July 2022 :

The values of fractions A, B, and C appear in the figure:

• Right edge: 1/6+ 1/2 = 1/6 + 3/6 = 4/6 = 2/3

• Left edge: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2

• Bottom edge: 1/3 + 1/2 = 2/6 + 3/6 = 5/6

Getting more from the problem...

The sum of the fractions at the vertices is 1/6 + 1/3 + 1/2 = 1/6 + 2/6 + 3/6 = 1.

The sum of the fractions in the boxes is 1/2 + 2/3 + 5/6 = 3/6 + 4/6 + 5/6 = 12/6 = 2.

The sum of the fractions in the boxes is twice the sum of the fractions at the vertices, because each box is the sum of two "vertex" fractions. Therefore, the sum of 3 "box" fractions is equal to the sum of 6 vertex fractions, which is twice the sum of the 3 vertex fractions.

#35: 10 July 2022 :

I know what 4/5 (“four 1/5s”) looks like, so I began by figuring out what 1/5 looks like. The first redrawn square divides the blue region into 4 regions, so each of those regions must equal 1/5. Since 5/5 = 1, I needed to figure out what 5/5 looks like. Any 5 regions that are the size of a divided blue region are equal to 1. In the second redrawn figure, 5 of these regions are shaded green. The green region represents 1.

#36: 11 July 2022 :

The sequence of images shows my strategy:

1. The bowl is 2/3 almonds and 1/3 cranberries.

2. The mixture is separated into halves.

3. Half of the mixture is removed.

4. The removed half is replaced with almonds.

As the last image illustrates, 5/6 of the new mixture is almonds.

#37: 12 July 2022 :

My solution is shown in the redrawn figure, which provides a visual representation of the equation.

#38: 13 July 2022 :

My solutions are shown in the redrawn models. There are also other ways to do these. For B, I could have shaded any 16 of the 20 partitioned regions.

#39: 14 July 2022 :

The equation is not true. 3 × 2/3 means 3 copies of 2/3 (“2 thirds”), which is 6/3 (“6 thirds”), or 2.

The misinterpretation is a reasonable one and it's pretty common. It's true that 6 out of the 9 regions are shaded, but the 9 regions do not represent the whole in this problem. If two green regions are 2/3, that means that each triangle is 3/3, or one whole.

#40: 15 July 2022 :

As shown in the figure, two-thirds of the 3 long rectangles is 6 small rectangles. Numerically, 2/3 × 3/5 = 6/15.

Swapping the order of the factors (i.e., 3/5 × 2/3) also yields a product of 6/15. [That's because multiplication is commutative.]

#41: 16 July 2022 :

The area of the rug is 3 1/3 square feet.

The redrawn figure contains the dimensions of each of the 6 parts of the rug. From left to right and top to bottom, the sum of their areas is:

1 + 1 + 1/2 + 1/3 + 1/3 + 1/6

= 2 + 1/2 + 2/3 + 1/6 = 2 + (3 + 4 + 1)/6

= 2 8/6 = 2 4/3 = 3 1/3

#42: 17 July 2022 :

The annotated figure shows that a large piece is the same as 3 small pieces. Since it takes 4 tbsp to make a large piece, it takes 4/3 tbsp to make a small piece. The question can be interpreted as, how many 4/3s are in 16? Or 16 ÷ 4/3 = ?

You could "flip and multiply" to find the answer, but there's no satis-fraction in that, so I'm going to find another way. Every 3 small pieces require 3 × 4/3 = 12/3 = 4 tbsp of sugar. So 6 pieces require 8 tbsp; 9 pieces require 12 tbsp; and 12 pieces require 16 tbsp, or 1 cup.

Or you could reason with the "unit" being 3 small pieces instead of 1 small piece: Since it takes 4 tbsp to make 3 small pieces, it takes 4 × 4 = 16 tbsp to make 3 × 4 = 12 small pieces.

12 small pieces can be made with 1 cup of sugar.

#43: 18 July 2022 :

Since 1 is equal to 3/3 (because there are three 1/3s in 1), then 2 is equal 6/3, and 3 is equal 9/3. The additional 2/3 (in 3 2/3) gives a total of 11/3. I've labeled the 11 regions in the redrawn figure. 11 regions are shaded, so each of them must represent 1/3.

We're asked to draw 1, which is 3/3 (“3 thirds”). My picture of 1 is in the blue box in the figure.

#44: 19 July 2022 :

The location of 2 1/3 /5 is shown on the redrawn number line.

Dividing the 1 yard into 5 parts lets you see where 1 fifth is. Then 2 fifths. And then 2 1/3 fifths.

#45: 20 July 2022 :

If the quotient of two numbers is 1, those numbers must be equal. That's because any number (other than zero) divided by itself is 1. If a and b are equal and their sum is 1, then they must be equal: a + b = 1 a + a = 1 (because a = b) 2a = 1, a = 1/2. Therefore, a = b = 1/2. And that means that their product, a × b, is equal to 1/2 × 1/2. Half of one half is one fourth, or 1/4.

#46: 21 July 2022 :

The figure contains my solutions. Those could be written as mixed fractions, too (e.g., 5 = 1 2/3). Or as decimals. Or percents!

I started by multiplying 5 by 2 to get the denominator of the fraction whose numerator is 3 × 2, or 6. Then I divided 10 by 3 to get the denominator of the fraction whose numerator is 6 ÷ 3, or 2. And so on...

#47: 22 July 2022 :

The figure contains my solutions. It's surprising that the difference between these unit fractions is equal to their product, right? Beware, though, this isn't true about every pair of unit fractions.

#48: 23 July 2022 :

A third and a half of a third is 1/3 + (1/2 × 1/3) = 1/3 + 1/6, which is 2/6 + 1/6 = 3/6, or a half.

One could also read this as "(a third and a half) of a third." In that case, you'd get (1/3 + 1/2) × 1/3 = (2/6 + 3/6) = 5/6 × 1/3 = 5/18. But that's no way to poetry.

#49: 24 July 2022 :

The solutions appear in the redrawn figure. In order, the equations indicate that T < S, T > A, H > T, and M < T.

How did I choose between 2/3 and 6/7 when I knew the fractions had to be less than 1? 6/7 is closer to 1 than 2/3 is, so 6/7 > 2/3: This is because 1/7 is smaller than 1/3. Since T is closer to S than it is to M, we want the fraction closer to 1 (6/7) to describe the relative locations of T and S (because they're closer together). And since T is farther from M than it is from S, we want the fraction farther from 1 (2/3) to describe the relative locations of T and M (because they're farther apart).

How did I choose between 5/4 and 9/8 when I knew the fractions had to be greater than 1? 9/8 is closer to 1 than 5/4 is, so 9/8 > 5/4: This is because 1/8 is smaller than 1/4. Since T is closer to H than it is to A, we want the fraction closer to 1 (9/8) to describe the relative locations of T and H (because they're closer together). And since T is farther from A than it is from H, we want the fraction farther from 1 (5/4) to describe the relative locations of T and A (because they're farther apart).

Phew. Good times.

#50: 25 July 2022 :

• |A| is 4/5 of |B|.

• |B| is 5/3 of |C|.

• |C| is 3/7 of |D|.

• |D| is 7/4 of |A|.

• Their product is 4/5 × 5/3 × 3/7 × 7/4 = (4 × 5 × 3 × 7)/(5 × 3 × 7 × 4) =(3 × 4 × 5 × 7)/(3 × 4 × 5 × 7) = 1.

The product of the four fractions is 1. Surprised?

#51: 26 July 2022 :

Theo, Vanessa, and Rudy's reasonings are all correct!

#52: 27 July 2022 :

If one person's share was 2/3 cheese pizza, that means that 2 cheese pizzas were shared among 3 people; so 8 cheese pizzas must have been shared among all 12 people (because 2/3 = 8/12). And if one person's share was 1/4 veggie pizza, that means that 1 veggie pizza was shared among 4 people; so 3 veggie pizzas must have been shared among all 12 people (because 1/4 = 3/12).

Therefore, 8 cheese pizzas and 3 veggie pizzas were ordered.

#53: 28 July 2022 :

Since there are 3 thirds in 1, multiplying the whole number (4) by the denominator (3) tells us that there are 12 thirds in 4. That's because every 1 contains 3 thirds, so 4 contains 3 × 4 = 12 thirds. So now we have 12 thirds, or 12/3.

The mixed number contains an additional 1 third. Adding that 1 third to the 12 thirds we already have gives us 13 thirds.

This figure confirms that 4 1/3 = 13/3. In other words, 4 wholes plus 1/3 is the same as 13 thirds.

#54: 29 July 2022 :

As hard as you try, you'll never make it to work.

This is because you'll always have 1/2 of the distance you've just traveled remaining: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256 ...

This is a variation on Zeno's "dichotomy paradox," which says you can never actually make it anywhere.

#55: 30 July 2022 :

Solving the equation that models this question will give me the copier setting I need: "What fraction of 4/5 of the original size (which is 100% or 1) produces a fraction that is 1/5 greater than the original size?" The fraction that is 1/5 greater than 1 is 6/5:

? × 4/5(1) = 6/5(1)

? × 4/5 = 6/5

So the product of some numerator and 4 is equal to 6. That's 6/4 (because 4 × 6/4 = 6). And the product of some denominator and 5 is equal to 5. That's 1. So the copier setting Roxy needs is 6/4 / 1 = 6/4: 6/4 × 4/5 = 6/5.

#56: 31 July 2022 :

From top left to bottom right, the four fractions are:

1 1/2 / 2 1/2 2 / 3 1/3

6/10 3/5

The four fractions are equivalent. That's because they all name the same shaded region (the numerator) and the same total area (the denominator). Doubling the numerator and denominator of the top left fraction produces the bottom right fraction, and tripling the numerator and denominator of the top right fraction produces the bottom left fraction.

#57: 1 August 2022 :

3 whole biscuits are equivalent to 3 × 4 = 12 quarter-biscuits. When 8 children share 12 quarter-biscuits, each kid gets 1 quarter-biscuit and a 1/2 from the remaining 4, for a total of 1 1/2 quarter-biscuits.

#58: 2 August 2022 :

All figures other than the one at the top right represent 4/9. The number line at the top right represents 6/9, not 4/9.

There are loads of WODB problems at wodb.ca

#59: 3 August 2022 :

1. As n gets bigger and bigger and bigger, 1/n gets smaller and smaller and smaller (e.g., 1/10, 1/100, 1/1000, ...), and closer and closer and closer to 0.

2. As n gets smaller and smaller and smaller, 1/n gets bigger and bigger and bigger (e.g., 1/(1/10) = 10; 1/(1/100) = 100; 1/(1/1000) = 1000, ...), and approaches infinity. [That's a way of saying it gets infinitely large, not that it approaches the number infinity. Infinity is a concept, not a number.]

#60: 4 August 2022 :

The figure on the right now shows my solution: 1/2 + 2/6 + 3/18 = 1.

#61: 5 August 2022 :

The figure now shows that each expression is equal to 4. There are 4 1/6s in 2/3, and 2/3 of 6 is 4.

So while it's NOT true that "dividing by a fraction is the same as multiplying by the reciprocal," it IS true that "dividing by a fraction is equivalent to (i.e., has the same value as) multiplying by the reciprocal."

#62: 6 August 2022 :

A. 1/2 of the figure is green. The reciprocal of 1/2 is 1/(1/2), or 2. That's the number of green regions that would fit in the figure.

B. 1/3 of the figure is purple. The reciprocal of 1/3 is 1/(1/3), or 3. That's the number of purple regions that would fit in the figure.

C. 1/6 of the figure is blue. The reciprocal of 1/6 is 1/(1/6), or 6. That's the number of blue regions that would fit in the figure.

D. If 1/1,000,000 of a figure is orange, then (loosely speaking) 1,000,000 orange pieces would fit into the figure.

#63: 7 August 2022 :

1. The part of the circular clock formed by the hour and minute hands at 4 o'clock is 1/3 of the entire circle.

2. The measure of the angle formed by the hour and minute hands is 1/3 of the 360 degrees in the circle, or 120 degrees.

#64: 8 August 2022 :

The length of arc BC is 4/25.13 of the circumference of the circle.

The reciprocal of 4/25.13 = 25.13/4, which is about 6.28. π is about 3.14, so that's 2π!

When the length of an arc (like BC) is equal to the radius (like AB), that arc is called a radian. This problem demonstrates that π (pi) isn’t just a number (~3.14159), it’s also a ratio that tells us how many radii (radiuses) fit on the circumference of any circle: π = C/2r. Wild, right? No matter the size of the circle, the circumference contains precisely 2π radii! This is what's shown in the revised figure: 2π (about 6.28) radii are numbered along the circumference of the circle.

#65: 9 August 2022 :

The expression, 100 ÷ 1/5, tells us how many 1/5-doses there are in 100 doses: 100 ÷ 1/5 = 100 × (1 ÷ 1/5) = 100 × (5) = 500 doses.

#66: 10 August 2022 :

Another 1/3 of the wall needs to be covered. If 2/3 of a wall is covered with 3/2 gallons of paint, then 1/3 of the wall was covered with 1/2 as much paint: 1/2 of 3/2 gallons is 3/4 of a gallon of paint.

#67: 11 August 2022 :

The correct matches are shown in this figure. All of the products are equal to 3/7 × 4/6 = 12/42. The center area model shows the product 1/42 of 12, which is also 12/42.

#68: 12 August 2022 :

I'm sure there are many, many correct answers to this problem. The one I have in mind is that the one that is unlike the others is on the lower left. In all of the others, there is an equal share of each color.

#69: 13 August 2022 :

The pink area is equal to the area of 3 small squares. The blue area is equal to the area of 13 small squares. So the pink area is 3/13 of the blue area.

#70: 14 August 2022 :

Here's the cost per can for each box:

\$16 for 12 cans is \$16/12 = \$1 4/12 = \$1 1/3

\$40 for 32 cans is \$40/32 = \$1 8/32 = \$1 1/4

\$48 for 40 cans is \$48/40 = \$1 8/40 = \$1 1/5

The smallest unit fraction has the largest denominator (i.e., 1/5 < 1/4 < 1/3), so in order from best value to worst, the boxes are size 40, size 32, and size 12. The 40-can box is the best value.

#71: 15 August 2022 :

The original array is 3/2 × 4/3. An array equal to 1 has dimensions 1 × 1. Since 2/2 (“two halves”) is equal to 1, and 3/3 (“three thirds”) is equal to 1, I can construct a 2/2 × 3/3 array that's equal to 1. The revised picture shows this 1 × 1 array within the larger array. Therefore, 1 is represented by 6 squares in the array.

Another strategy is to first find the product, 3/2 × 4/3. That's equal to 12/6, or 2. Since 2 is represented by 12 squares, then 1 is represented by half as many squares, or 6 squares.

#72: 16 August 2022 :

The distance between each mark on the track is 1/3 of a meter, so each lap is 5/3 (“5 thirds”) meters. So the question, How many laps are there in an ant race? becomes How many 5/3 meters are in 10 meters? Or, what is 10 ÷ 5/3?

Ever find common denominators when dividing? We usually only do that when adding or subtracting, but it turns out to be really useful here. Check this out: 10 ÷ 5/3 = 30/3 ÷ 5/3. There are 6 copies of 5/3 in 30/3, so 30/3 ÷ 5/3 = 6.

An ant race is 6 laps around this track.

#73: 17 August 2022 :

The yellow regions are the following fractions of a whole:

Triangle: 1/6 × 1/10 = 1/60

Square: 1/4 × 1/15 = 1/60

Pentagon: 1/5 × 1/12 = 1/60

Hexagon: 1/3 × 1/20 = 1/60

#74: 18 August 2022 :

8 out of 12 is the right answer. 3/6 ("3 out of 6") and 5/6 ("5 out of 6") are ratios, and sometimes ratios don't behave like part-whole fractions. It's true that Brittany made 3 out of 6 shots on the first day and 5 out of 6 shots on the second day, but she didn't make 8 out of 6 shots in all (the sum of two fractions). That wouldn't be possible. She made 8 out of 12 shots (the sum of two ratios). The revised picture shows each ratio and what ratio arithmetic looks like to avoid the confusion between fraction arithmetic (which requires a common denominator) and ratio arithmetic (which doesn't).

#75: 19 August 2022 :

The blue region is 1/2 of the area of this design. The blue region is also 1/40 of the area of the quilt. If 1/2 of a panel is 1/40 of the quilt, then (doubling both) 1 panel must be 1/20 of the quilt. If each panel is 1/20 of the quilt, then the quilt contains 20 panels.

Another way to solve this is as follows: If 1/2 a panel is 1/40 of the quilt, then the panel must be 1/20 of the quilt. That's because 1/2 of 1/20 is 1/40. In other words, 1/2 of 1/20 of the quilt is 1/40 of the quilt. If each panel is 1/20 of the quilt, then the quilt contains 20 panels.

#76: 20 August 2022 :

The fraction that completes the equation is: output = 2/3 × input

This is the fraction as operator meaning of a fraction.

#77: 21 August 2022 :

As shown in the redrawn figure, Emma's share is 3/18 of the cake and Mika's share is 2/9, or 4/18. So the question is, "3/18 is what fraction of 4/18?" That's equivalent to the question, "3 parts is what fraction of 4 parts?" And the answer to that question is 3/4.

#78: 22 August 2022 :

The area of the blue region is 1/2(3 × 2), or 3. The area of the green region is 1/2(1 × 6), or 3. [The white and yellow regions have area 3, as well!] The area of the flag is 6 × 3, or 18, so the area that is neither blue nor green is 18 - 3 - 3, or 12. That means 12/18, or 2/3, of the flag of Seychelles is neither blue nor green.

#79: 23 August 2022 :

"1/5 ÷ 2/3" means, "How many 2/3s are in 1/5?" Since 1/3s and 1/5s are different sizes, I've found equivalent fractions that are the same size: 1/5 = 3/15 and 2/3 = 10/15. So now the question is, "How many 10/15s are in 3/15?" Or "How much of that 10-square-sized pink region fits into that 3-square-sized blue region?" Less than 1 of them, right? As shown in the redrawn figure, only 3/10 of the pink region fits into the blue region. So 1/5 ÷ 2/3 = 3/10.

#80: 24 August 2022 :

Fill in the blanks: If there are n equal-sized pieces in a whole, then each piece is 1/n of the whole.

Now find two fractions whose product is 1: n × 1/n = 1.

Two fractions are multiplicative inverses of each other if their product is 1.

Have a great day! :)

#81: 25 August 2022 :

• 30 ÷ 1/2 = 60: Each whole contains 2 halves, so 30 wholes contains 30 × 2 = 60 halves.

• 20 ÷ 1/3 = 60: Each whole contains 3 thirds, so 20 wholes contains 20 × 3 = 60 thirds.

• 15 ÷ 1/4 = 60: Each whole contains 4 fourths, so 15 wholes contains 15 × 4 = 60 fourths.

• 12 ÷ 1/5 = 60: Each whole contains 5 fifths, so 12 wholes contains 12 × 5 = 60 fifths.

#82: 26 August 2022 :

As this "solution" figure shows, 1/4 of the mixture contains orange candy. 1/3 of the mixture that's not orange contains lemon candy. 1/2 of the mixture that's neither orange nor lemon contains strawberry candy. And 1/2 of the mixture that's neither, orange, lemon, or strawberry contains grape candy.

What remains is 1/2 of a 1/4, or 1/8, of the mixture. That fraction of the mixture could be blueberry.

#83: 27 August 2022 :

A wiser way to do this is to determine how much of the area is unshaded: That's 1/4 (for the white trapezoid) + 1/4 of 1/6, or 1/24 (for the white small triangle): 1/4 + 1/24 = 6/24 + 1/24 = 7/24. That means the shaded region is 1 minus 7/24, which leaves 17/24.

A more complicated way might be more fun, though, and that's to determine how much of the area of the top hexagon is shaded using the other hexagons, as the question called for. The top hexagon contains:

1/3 of a red 1/2, or 1/6; 1/2 of a blue 1/3, or 1/6; 1 yellow 1/4; and 3 green 1/24s. That's a total shaded area of 1/6 + 1/6 + 1/4 + 3/24 = 4/24 + 4/24 + 6/24 + 3/24 = 17/24.

#84: 28 August 2022 :

This figure shows that 3/6 = 12/24, and x/4 = 6x/24. Since the denominators of these ratios are equal, then their numerators must be equal (Why is that?). That's what the red arrow points to.

So what did we multiply each ratio by? The answer is that each ratio was multiplied by a fraction equal to 1 and of the form n/n, where n is the denominator of the other ratio. Take a look at the figure to see what I mean.

#85: 29 August 2022 :

a can be any whole number; b is equal to 1:

• If a pieces of Albert's Fruit Chews are shared among b = 1 person, then that person gets a/b = a/1 = a pieces of candy.

• If there are a pieces of Albert's Fruit Chews in each bag and I have b = 1 bag, then I have a pieces of candy.

#86: 30 August 2022 :

1/2 ÷ 1/6 = 3, since three 1/6s fit into 1/2. And 1/6 ÷ 1/2 = 1/3, since only 1/3 of a 1/2 fits into 1/6. So this complex fraction is equivalent to the proper fraction 3/(1/3). 3 ÷ 1/3 is equal to 9, because each 1 contains three 1/3s. That means 3 contains nine 1/3s. So this complex fraction is equal to 9.

#87: 31 August 2022 :

From left to right and top to bottom, this figure shows just one of the ways the square can be divided into two 1/2s, three 1/3s, six 1/6s, and nine 1/9s.

#88: 1 September 2022 :

• Since a < b, then 1/a > 1/b [because the greater fraction has the smaller denominator].

• Then 3 + 1/a > 3 + 1/b [because we just added the same amount to both fractions].

• Then 1/(3 + 1/a) < 1/(3 + 1/b) [because the greater fraction has the smaller denominator].

• Then 2 + 1/(3 + 1/a) < 2 + 1/(3 + 1/b) [because we just added the same amount to both fractions].

• Then 1/[2 + 1/(3 + 1/a)] > 1/[2 + 1/(3 + 1/b)] [because the greater fraction has the smaller denominator].

#89: 2 September 2022 :

• The red rod is 2/7 of the black rod.

• The yellow rod is 5/2 or 2 1/2 of the red rod.

• If the white rod is 1/9, the sum of three dark green rods is equal to 2. Each dark green rod is 6/9, so three of them are equal to 18/9, or 2.

#90: 3 September 2022 :

One-fourth of two-thirds of the orange splat's area is equal to two-thirds of three-fourths of the blue splat's area:

--> 1/4 × 2/3 of orange = 2/3 × 3/4 of blue

--> 2/12 orange = 6/12 blue

--> 1/6 orange = 1/2 blue

If I take 6 times that amount of orange and 6 times that amount of blue (to maintain the equality), I learn that 6/6, or 1 orange splat = 6/2, or 3 blue splats. So 1 orange splat is 3 times the size of 1 blue splat.