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Subtracting natural numbers
How many
\latex{ km } are left?
\latex{ km } are left?
Dan and his friends go hiking to visit a castle found \latex{ 7\,km } from the village. After \latex{ 3\,km }, they stop at a well to drink. How many \latex{ kilometres } do they still have to walk to the castle?
\latex{7-3 = 4}
- subtrahend =
difference
minuend
The properties of subtraction
What happens when the minuend and subtrahend are switched?
\latex{7-3 = 4}; \latex{ 3-7} is not a natural number, so
\latex{7-3 = 4}; \latex{ 3-7} is not a natural number, so
\latex{7-3\neq 3-7.}
The difference can change when the minuend and subtrahend are switched.
\latex{a-b \neq b-a}
When subtracting several numbers, use brackets to indicate the order of operations.
Example
A florist had \latex{ 14 } carnations and \latex{ 14 } roses. The first customer bought \latex{ 5 } carnations, the second chose \latex{ 2 } of the remaining carnations. The third customer wanted \latex{ 5 } roses, but ended up giving \latex{ 2 } back.
Are the same number of roses and carnations left?
Solution
Carnations: \latex{(14 - 5) - 2 = 9 - 2 = 7.}
Roses: \latex{14 - (5 - 2) = 14 - 3 = 11.}
There are \latex{7} carnations and \latex{11} roses left.
\latex{(14-5)-2\neq 14 - (5-2)}
Changing the position of the brackets might modify the result of a subtraction.
\latex{(a-b)-c\neq a - (b-c)}
The relationship between addition and subtraction
1. Checking an addition:
Subtraction is used to check an addition.
2. Checking a subtraction:
Both addition and subtraction can be used to check a subtraction.
\latex{3+4=7}
\latex{7-4=3}
\latex{7-3=4}
\latex{7-4=3}
\latex{3+4=7}
\latex{7-3=4}
Changes in the difference
Observe how the difference changes when the minuend and subtrahend are changed.
\latex{7-4=3}
\latex{7 - 4 = 3}
increased
by \latex{ 2 }
by \latex{ 2 }
decreased
by \latex{ 2 }
by \latex{ 2 }
The minuend is changed
\latex{9 - 4 = 5}
If the minuend is increased by \latex{ 2 },
the difference also increases by \latex{ 2 }.
the difference also increases by \latex{ 2 }.
If the minuend is decreased by \latex{ 2 },
the difference also decreases by \latex{ 2 }.
the difference also decreases by \latex{ 2 }.
increased
by \latex{ 2 }
by \latex{ 2 }
(the subtrahend remains unchanged)
decreased
by \latex{ 2 }
by \latex{ 2 }
\latex{5 - 4 = 1}
\latex{7 - 4 = 3}
\latex{7 - 4 = 3}
increased
by \latex{ 2 }
by \latex{ 2 }
decreased
by \latex{ 2 }
by \latex{ 2 }
The subtrahend
is changed
is changed
\latex{7 - 6 = 1}
If the subtrahend is increased by \latex{ 2 },
the difference decreases by \latex{ 2 }.
the difference decreases by \latex{ 2 }.
If the subtrahend is decreased by \latex{ 2 },
the difference increases by \latex{ 2 }.
the difference increases by \latex{ 2 }.
decreased
by \latex{ 2 }
by \latex{ 2 }
(the minuend remains unchanged)
increased
by \latex{ 2 }
by \latex{ 2 }
\latex{7 - 2 = 5}
\latex{7 - 4 = 3}
\latex{7 - 4 = 3}
increased
by \latex{ 2 }
by \latex{ 2 }
decreased
by \latex{ 2 }
by \latex{ 2 }
Both the minuend and subtrahend are changed by the same number
\latex{9 - 6 = 3}
If both the minuend and subtrahend are
increased by \latex{ 2 }, the difference remains unchanged.
increased by \latex{ 2 }, the difference remains unchanged.
If both the minuend and subtrahend are
decreased by \latex{ 2 }, the difference remains unchanged.
decreased by \latex{ 2 }, the difference remains unchanged.
increased
by \latex{ 2 }
by \latex{ 2 }
decreased
by \latex{ 2 }
by \latex{ 2 }
\latex{5 - 2 = 3}
\latex{7 - 4 = 3}
unchanged
unchanged
The difference remains unchanged if both the minuend and subtrahend are changed by the same amount.
Exercises
Make an estimation before solving exercises 1–5. , then check your answers.
{{exercise_number}}. Perform the following subtractions in your head.
a) \latex{6,750-1,750}
b) \latex{4,000-2,500}
c) \latex{2,300-900}
d) \latex{35,000-26,000}
e) \latex{1,910-1,010}
f) \latex{1,910-910}
g) \latex{11,910-10,910}
h) \latex{11,910-11,010}
i) \latex{11,910-10,900}
{{exercise_number}}. What is the difference when subtracting
a) six thousand and four from nine thousand seven hundred and four;
b) five thousand and ninety-two from fifty thousand nine hundred and twenty;
c) one hundred twenty thousand three hundred and fifty-five from one million?
{{exercise_number}}. What is the difference if
a) the minuend is greater than the subtrahend by \latex{ 172 };
b) the subtrahend is \latex{ 200 } and the minuend is \latex{ 3 } times larger than the subtrahend;
c) the minuend is \latex{ 57,041 }, and the subtrahend is \latex{ 9,999 };
d) the subtrahend is \latex{ 236 } smaller than the minuend?
{{exercise_number}}. How much change do you get from €\latex{ 25 } if you spend €\latex{ 8 }? The following day, you spend €\latex{ 3 } less than the previous day. How much change do you get from €\latex{ 25 }?
{{exercise_number}}. In a nursery, \latex{ 2,340 } pepper and tomato plants are grown in total. There are \latex{ 1,150 } tomato plants.
a) How many pepper plants are there?
b) Which plant is grown in larger numbers? By how much?
{{exercise_number}}. Agnes has €\latex{ 25 } because she got €\latex{ 3 } yesterday. How much money did she have the day before yesterday if she did not spend any money?
{{exercise_number}}. What is the difference of the smallest five-digit and the largest three-digit natural numbers?
{{exercise_number}}. Use the digits \latex{ 2 }; \latex{ 3 }; \latex{ 5 } and \latex{ 7 } to make all the possible four-digit numbers where each digit is used only once and \latex{ 7 } is in the hundreds place.
{{exercise_number}}. Use the digits \latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 } and \latex{ 5 } to make all the possible six-digit numbers where \latex{ 1 } is in the units place, \latex{ 2 } is in the tens place and \latex{ 3 } is in the thousands place. Every digit can be used only once.
a) What is the difference of the largest and the smallest numbers?
b ) What is the difference of the sum of the two largest and the sum of the two smallest numbers?
{{exercise_number}}. Tom is \latex{ 5 } years older than Frank, who is \latex{ 14 } years old. What will the difference between their ages be in seven years?
{{exercise_number}}. How much was added to the minuend if the subtrahend was decreased by \latex{ 5 } and the difference increased by \latex{ 15 }?
{{exercise_number}}.
a) There were \latex{ 39 } tents and \latex{ 15 } cabins at a camp. How many more tents are there than cabins?
b) The other camp had \latex{ 4 } less tents and \latex{ 6 } more cabins. How many more tents are there in this camp than cabins?
c) How did the difference change?
b) The other camp had \latex{ 4 } less tents and \latex{ 6 } more cabins. How many more tents are there in this camp than cabins?
c) How did the difference change?
{{exercise_number}}. After performing the subtraction, the difference is \latex{ 12,457 }. What will the difference be
a) if the minuend is increased by \latex{ 140 };
b) if the subtrahend is decreased by \latex{ 43 };
c) if both the minuend and subtrahend are decreased by \latex{ 200 };
d) if both the minuend and the subtrahend are increased by \latex{ 4,973 };
e) if the minuend is increased by \latex{ 137 } and the subtrahend is decreased by \latex{ 163 }?
{{exercise_number}}.
a) What is the smallest and largest five-digit natural number? What is their difference?
b) How does the difference change if the largest five-digit number is increased by \latex{ 100 } and the smallest is decreased by \latex{ 100 }?
b) How does the difference change if the largest five-digit number is increased by \latex{ 100 } and the smallest is decreased by \latex{ 100 }?
{{exercise_number}}. Dorothy, Ben and Claire live in the same street as the school. Dorothy lives \latex{ 1,327 }, Ben lives \latex{ 2,433 }, and Claire lives \latex{ 975\,m } from the school. How far do the children live from each other if the street is straight?
{{exercise_number}}. Perform the subtractions.
a) \latex{9,675-7,520}
b) \latex{9,675-5,897}
c) \latex{10,100-8,654}
d) \latex{82,916-63,818}
e) \latex{57,976-15,673}
f) \latex{74,302-63,818}
Quiz
Peaches were put in \latex{ 7 } boxes. The first one contained a certain amount of peaches, while the other boxes had \latex{ 3 } more peaches than the previous one. As a result, there were twice as many peaches in the last box than in the first one. How many peaches were put in each box?
