Arithmetic, Mental Math and Unit Conversion | Operations, Geometry and Ratios, Set 2
Problem 1
What is the value of 48 ÷ (6 + 2) × 5?
View explanation
First work out the bracket: 6 + 2 = 8. Then division and multiplication proceed from the left, giving 48 ÷ 8 = 6 and 6 × 5 = 30. The answer 3 comes from pulling the × 5 inside the bracket, and 18 from ignoring the bracket and computing 48 ÷ 6 + 2 × 5.
Problem 2
You read two-fifths of a book in the morning and one-third of it in the afternoon. Taking the two together, how much of the book have you read?
View explanation
Both fractions are of the whole book, so add two-fifths and one-third. Putting them over 15 gives two-fifths = six-fifteenths and one-third = five-fifteenths, which total eleven-fifteenths. Three-eighths and five-eighths come from adding the denominators together, which is not how fractions are added.
Problem 3
What is the value of 3.6 × 0.25?
View explanation
0.25 is one quarter, so 3.6 × 0.25 is the same as 3.6 ÷ 4 = 0.9. You can also check it by computing 36 × 25 = 900 as whole numbers: the two factors have three decimal places between them, so the product is 0.900 = 0.9. The answers 9 and 0.09 misplace the decimal point.
Problem 4
An L-shaped plot is formed by removing a 3 m by 2 m rectangle from the corner of a plot 8 m long and 6 m wide. What is the area of the remaining land, in square meters?
View explanation
The outer rectangle is 8 × 6 = 48 m² and the piece removed is 3 × 2 = 6 m², so what remains is 48 − 6 = 42 m². The figure 48 m² is the area before anything was cut out, and 54 m² adds the removed piece instead of subtracting it. The figure 36 m² is (8 − 2) × (6 − 3), subtracting the side lengths first, which does not describe the L-shape.
Problem 5
There are two rectangular containers with internal dimensions 30 cm × 20 cm × 15 cm and 20 cm × 15 cm × 10 cm. Filled to the brim, how many liters of water do they hold in total? Take 1,000 cm³ = 1 L.
View explanation
The first has a volume of 30 × 20 × 15 = 9,000 cm³ and the second 20 × 15 × 10 = 3,000 cm³. The total is 12,000 cm³, and since 1,000 cm³ = 1 L, that is 12 L. The answers 9 L and 3 L give only one container each, and 120 L is out by a factor of ten in the conversion from cm³ to L.
Problem 6
Four people who took the morning training averaged 72 on the test, and six who took it in the afternoon averaged 82. What is the average for all ten?
View explanation
The four in the morning total 72 × 4 = 288 and the six in the afternoon 82 × 6 = 492, giving 780 overall. Divided among ten people that is 780 ÷ 10 = 78. The answer 77 simply averages 72 and 82, which does not give the overall average because the groups differ in size.
Problem 7
There are 64 cards in total, red and blue, in the ratio red : blue = 3 : 5. How many blue cards are there?
View explanation
The parts of the ratio total 3 + 5 = 8, so one part is 64 ÷ 8 = 8 cards. Blue is five parts, giving 8 × 5 = 40. The answer 24 is the three parts of red, 8 is a single part, and 32 simply halves the total without using the ratio.
Problem 8
As of July 2026, assume the reduced tax rate of 8% applies to takeaway bento and the standard rate of 10% to a ceramic plate. If you buy one bento at 1,000 yen before tax and one plate at 2,000 yen before tax, what is the total including tax? No fractions of a yen arise in the tax amounts.
View explanation
The bento is 1,000 × 1.08 = 1,080 yen and the plate 2,000 × 1.10 = 2,200 yen, so the total including tax is 1,080 + 2,200 = 3,280 yen. The figure 3,240 applies 8% to both and 3,300 applies 10% to both, failing to separate the two rates. The rates are those in National Tax Agency materials as of July 19, 2026.
Problem 9
You want to buy rice of the same quality in whichever bag is cheapest per 100 g. Which of the following is cheapest?
View explanation
Per 100 g, the 240 g bag at 360 yen works out at 150 yen, the 350 g bag at 490 yen at 140 yen, the 500 g bag at 750 yen at 150 yen and the 600 g bag at 900 yen also at 150 yen. The 350 g bag at 490 yen is therefore the cheapest. By total price alone the 240 g bag looks cheapest, but the quantities differ, so you have to compare on the same per-100 g basis.
Problem 10
What is 0.35 m² expressed in square centimeters? Take 1 m = 100 cm.
View explanation
For area both dimensions are converted, so 1 m² = 100 cm × 100 cm = 10,000 cm². Hence 0.35 × 10,000 = 3,500 cm². The answer 350 cm² multiplies by 1,000 and 35 cm² by 100, as though converting a length, neither of which reflects the change in a squared unit.