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Keeping the balance
If you place equal masses in both pans of a two-pan scale, it is balanced. You can keep the scale balanced by putting equal masses into the pans or removing equal masses from them. By repeating this method, you can eventually have the unknown mass in one pan and a known mass in the other. At that point, you will be able to determine the value of the unknown mass as it will be equal to the mass in the other pan.
Example 1
A two-pan scale is balanced. There is a pear, a weight of \latex{ 500 \;grams } and a weight of \latex{ 250 \;grams } in one of the pans. The other pan contains a weight of \latex{ 1,000 \;grams }.
What is the mass of the pear?
Solution
If you remove \latex{ 500 + 250 = 750 \;grams } from both pans, the scale remains balanced. The pear will then be in balance with a weight of \latex{ 1,000 - 750 = 250 \;grams }.
Therefore, the mass of the pear can be determined.
Check:
If the mass of the pear is \latex{ 250 \;grams }, then there were \latex{ 500 + 250 + 250 = 1,000\; grams } in each of the pans, which means that the scale was balanced.
Answer:
The mass of the pear is \latex{ 250 \;grams }.
If a two-pan scale is balanced, the balance will be maintained if the same amount of mass is added to or removed from both of the pans.
Example 2
The children traded the chocolates they received for Easter. Erick got three bunnies and one sheep for one egg, and Bruno got four bunnies for two sheep. How many bunnies is one egg worth?
Solution
If two sheep are worth four bunnies, then one sheep is worth half of that, which is two bunnies.
Therefore, three bunnies and one sheep are worth \latex{ 3 + 2 = 5 } bunnies. Thus, one egg is worth five bunnies.
Check:
If one egg is worth five bunnies, and Eric got three bunnies and one sheep for one egg, then one sheep is worth two bunnies. Two sheep are worth \latex{ 2 × 2 = 4 } bunnies. Thus, both requirements of the exercise are fulfilled.
Answer:
One egg is worth five bunnies.
If a two-pan scale is balanced, the balance will be maintained if you halve the original mass in both pans.
If a two-pan scale is balanced, the balance will be maintained if you double the original mass in both pans.
Exercises
{{exercise_number}}. A store sells honey in two types of packages. Based on the image below, which type is heavier?
{{exercise_number}}. One of the pans of a two-pan scale contains a package and a weight of \latex{250 \;grams }. The other pan has a weight of \latex{ 5 \;kg } in it. The scale is balanced. What is the mass of the package?
{{exercise_number}}. There are three apples of equal mass and a weight of \latex{ 100\, grams } in a pan of a two-pan scale. The other pan has a weight of \latex{ 1\; kg } in it. The scale is balanced. How many \latex{ grams } is one apple?
{{exercise_number}}. The mass of three bags of flour is \latex{ 5\; kg } less than the mass of two bags of flour and a weight of \latex{ 25\; kg }. How many \latex{ kilograms } is the mass of one bag of flour if all bags have equal mass?
{{exercise_number}}. You bought five kiwis and two mangos in a store for €\latex{ 25.5 }. If one mango costs six times as much as one kiwi, then how much does one mango and one kiwi cost separately?
{{exercise_number}}. If three apples and two oranges cost \latex{ 20 }¢ more than two apples and three oranges, how much more does one apple cost than one orange?
{{exercise_number}}. Three students are travelling to another city to take part in a math competition. The total mass of their bags is \latex{ 65 \;kg }. The mass of the two heavier bags is \latex{ 52\; kg }, and the mass of the two lighter ones is \latex{ 38\; kg }. What is the mass of the lightest bag? Can you determine the mass of the middle one?
{{exercise_number}}. The sum of three numbers is \latex{ 2,002 }. The average of the two smaller numbers is \latex{ 485 }, and the average of the two larger numbers is \latex{ 771 }. What are the three numbers?
{{exercise_number}}. A puppet maker makes three types of puppets: lions, tigers, and monkeys. He uses the same amount of material to make two tigers as to make eight monkeys, and he uses the same amount of material to make one tiger and two monkeys as to make two lions. How many times more material does he need to make one lion than to make one monkey?
{{exercise_number}}. The luggage of the first-year students in a wizard school is shipped by boat. Four wooden chests, a larger one and three smaller ones that are equal in mass, were put into a boat.
The larger chest is \latex{ 30 \;kg } heavier than one of the smaller chests, and the combined mass of the three smaller chests is \latex{ 30 \;kg } heavier than the larger chest. Determine the masses of the chests in \latex{ kilograms. }
{{exercise_number}}. Four girls bought flowers. Anne bought five carnations for €\latex{ 10 }, Blair bought one rose, one carnation and two irises for €\latex{ 18 }, and Clara bought one rose and three irises for €\latex{ 20 }.
How much did Dora pay if her bouquet was made of two roses, one carnation and two irises?
{{exercise_number}}. The beams of the scale shown in the image are balanced. The horizontal rods are suspended at their midpoint, and their two sides are the beams of the scale. The same figures are equal in mass. The mass of the green square is \latex{ 1 \;g }.
What is the mass of the other figures?
a)
b)
Quiz
Four squirrels collected a total of \latex{ 2,001 } nuts to prepare for winter. Each squirrel collected more than \latex{ 100 } nuts. The oldest squirrel collected more nuts than any of the others. The second-oldest and third-oldest squirrels collected \latex{ 1,266 } nuts together. How many nuts did the squirrels collect separately?
